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Electromagnetic coupling between the fluid core and its solid neighbours Dumberry, Mathieu 1998

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Electromagnetic Coupling Between the Fluid Core and its Solid Neighbours By Mathieu Dumberry B.Sc. (Physics), Universite de Sherbrooke  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E O F M A S T E R OF SCIENCE  in T H E FACULTY OF G R A D U A T E STUDIES D E P A R T M E N T OF EARTH A N D O C E A N 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 i n partial fulfillment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying 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 m y written permission.  E a r t h and Ocean Sciences T h e University of B r i t i s h Columbia 129-2219 M a i n M a U Vancouver, B C , Canada V 6 T 1Z4  Date:  11  Abstract A t low-frequency, the nearly geostrophic force balance i n the fluid core constrains axisymmetric fluid motions to be purely azimuthal and independent of position along the rotation axis. F l u i d motions can thus be described by a set of concentric rigid cylinders, which are free to rotate about their common axis. W h e n 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 i n the length of day by transferring angular momentum to the mantle and the inner core. T h e 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 i n the fluid near the boundary. However, it is not valid at the inner core boundary ( I C B ) where higher field strengths are likely to perturb the geostrophic balance. In this case, the Lorentz force has to be retained i n the m o m e n t u m balance. A complete analytical solution is given for the influence of Lorentz forces i n the core. T h e model problem involves a conducting and rotating fluid between two plane conducting boundaries i n 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.  At  decade periods, there is insufficient inertia to disrupt rigid rotations. However, for annual fluctuations,  departures i n the fluid velocity from rigid rotations are significant, which  implies that the Lorentz force can not be neglected i n the dynamics of the fluid core when calculating the electromagnetic coupling at the I C B .  Ill  Contents Abstract  ii  Table of contents  iii  List of figures  v  List of tables  vii  Acknowledgments . . .  viii  Introduction  1  Formulation of the problem  13  2.1  Geometry and basic equations of the model  2.2  F o r m of the solution of the equations  2.3  Boundary conditions  13 .  24  Single-plate problem 3.1  Introduction  3.2  Solution of the problem  3.3  Results and discussion  20  29 ' .•  29 30  . .  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 . . .  4.2  Solution of the problem  51  4.3  Results and discussion  57  4.3.1  Physical description of the solution  4.3.2  Transfer of angular momentum  ..  .  51  58 61  Table of contents  iv  4.3.3  Relative magnetic stresses at each boundary  63  4.3.4  R i g i d 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 B  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  z  column  Rigid rotations from a concentrated torque  86  93  6.1  Introduction  93  6.2  M o d e l 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  R i g i d rotations i n 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 i n the fluid and the solid at frequency to = ft/10  and magnetic field B  4  3.3  and magnetic field B  2  47  Solution for the scalar potentials i n the fluid and the solid at frequency and magnetic field B  6  = 5G  Z  48  Solution for the scalar potentials i n the fluid and the solid at frequency u> = ft/10  and magnetic field B  4  3.6  = 5G  z  uj = ft/10 3.5  46  Solution for the scalar potentials i n the fluid and the solid at frequency u = ft/10  3.4  = 5G  z  = 0.5G  2  49  Solution for the scalar potentials i n the fluid and the solid at frequency u> = ft/10  4  and magnetic field B  z  = 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 i n the fluid at a magnetic field B  Z  4.4  = 5G  68  Solution for the scalar potentials of the magnetic field i n the fluid at a magnetic field B  z  4.5  Z  = 35G  70  Solution for the scalar potentials of the magnetic field i n the fluid at a magnetic field B  z  4.7  = 35 G  71  Solution for the scalar potentials of the velocity i n the fluid at a magnetic field B  z  4.8  69  Solution for the scalar potentials of the velocity i n the fluid at a magnetic field B  4.6  = 5G  = 100 G  72  Solution for the scalar potentials of the magnetic field i n the fluid at a magnetic field B  z  = 100 G  73  List of figures 5.1  vi  Transfer of angular momentum to the fluid as a function of magnetic field strength at a period of 200 days  5.2  88  Transfer of angular momentum to the fluid as a function of magnetic field strength at a period of 400 days  5.3  88  Transfer of angular momentum to the fluid as a function of the period of oscillation for B = 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  z  velocities (part I) 5.9  92  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  6.3  Solution for the scalar potentials of the velocity at a magnetic field B = 5 C7.107  6.4  Solution for the scalar potentials of the magnetic field at a magnetic field  in the disc  100 z  B = 5G  108  Z  6.5  Solution for the scalar potentials of the velocity at a magnetic field B = 50 G. 109  6.6  Solution for the scalar potentials of the magnetic field at a magnetic field  z  B. = 50(3 z  6.7  Solution for the scalar potentials of the velocity, when the coupling at the fluid-solid boundaries is included  6.8  110  Ill  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 i n calculation of the electromagnetic coupling between the fluid and a single plate  4.1  Parameters used i n calculation of the electromagnetic coupling between a cylinder of fluid and two bounding plates  5.1  77  Parameters used i n calculating the torque on the fluid column due to a concentrated force  6.2  58  Parameters used i n comparison of the angular momentum transfer between the rigid rotation model and the more complete model  6.1  35  103  Parameters used i n calculating the torque on the fluid column due to combined effects of electromagnetic coupling w i t h 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 m y life so enjoyable that I have to use single spacing to fit it all on one page. First of all, a million thanks to m y advisor Bruce Buffett, for his incredible patience and extraordinary mentorship. I've learned so much during the countless hours I spent i n his office, appreciating his approach to science and to life i n general. Y o u were a dream advisor, as you gave me the perfect balance between guidance and freedom. Y o u have been a model that I have little hope to live up to. Thanks to everyone i n the geophysics building, starting w i t h 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 K a r e n Wijns for being the child you are and the child you will always be. Thanks T i a h Goldstein, you gave me a glimpse of life outside the department. Thanks Colin Farquharson for your calm, your quietness, and for the boots! Thanks Dave Hildes, for your constant enthusiasm and joviality. Thanks Shawn M a r s h a l l for being an inspiration i n many domains. Thanks Gwenn Flowers,. for your radiating happiness and great companionship. M a n y thanks K e v i n K i n g d o n , for your reassuring presence. Y o u won the contest of the person who disturbed me the most often i n m y work, and believe me, it was always appreciated. I find it impossible to dissociate you from my stay i n 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 i n many ways. In the "everyday life" category, thanks to Calhoun's for letting me use their tables as m y 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 m y delirious brain. I don't think I've ever walked i n 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 i n the reading room (always comfortable after 2 a.m.!), to Wrang and Inverse (my two most intimate companions i n the department!), to my commuting partner "Hard Rock" and m y riding partner V i t a l i , to L a Bicicletta, to Hollyburn, to the ski-car, to the $5 ladies-series, to O a t m e a l C i t y and Friday afternoon soccer, to D r . 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 E a r t h over the past three hundred years show that the length of the day (or L O D ) is not constant [McCarthy and Babcock 1986]. T h e E a r t h as a whole is exchanging angular momentum with the moon by means of tidal friction. T h e energy lost by the E a r t h from tidal dissipation reduces its angular velocity. T h e time the E a r t h 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 M o r r i s o n 1984]. T h e 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.)]. T w o 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 E a r t h (or the mantle) is not constant. Hence, the coupling between the mantle and the other constituents of the E a r t h permits an exchange of angular momentum. T h e 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 i n the atmosphere account for almost all of the changes i n mantle's angular m o m e n t u m at annual periods. T h e 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 i n the angular m o m e n t u m exchange [Chao 1994].  Chapter 1: Introduction  2  ISSO  1985  D a t e  ISSO  1S95  ( 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 i n the angular momentum fluctuations i n the atmosphere to explain the long period oscillation of the L O D variations.  Since no known exterior  torque acts on the E a r t h at such frequency, the transfer of angular momentum can only be explained by internal coupling between the core and the mantle. T h e flow i n the core is not directly observable, so we must rely on theoretical models to assess the angular m o m e n t u m carried by the core. At time scales much longer than the period of rotation of the E a r t h , and i n the absence of magnetic forces, the flow i n the outer liquid core is geostrophic. T h a t is, the Coriolis force is balanced by the pressure gradients. T h e 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 i n 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 i n 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). W h e n the magnetic force (often referred as the Lorentz force) is reinstated, the basic flow i n 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: T h e Proudman-Taylor theorem applied to the Earth's fluid core restricts the motion to rigid rotations aligned w i t h the rotation axis.  Jault et al. [1988] and Jackson et al. [1993] have computed the angular m o m e n t u m carried by the core at decade time scale using this model for the flow i n 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 Earth. T h e decade time scale of these variations is much shorter than the diffusion time for the magnetic field i n the core, which is on the order of 10 years. 4  T h e 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 will follow their motion. This is known as Alfven's frozen-flux hypothesis. The magnetic field i n 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 i n the magnetic field at the surface of the E a r t h . T h e 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 electromagnetic. The gravitational coupling between the fluid and the mantle arises from aspherical density variations due to convective fluctuations [Jault and Le Mouel 1989]. This coupling 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 coupling 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  5  Chapter 1: Introduction  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 w i t h their motion. A shear i n velocity between two conductors will stretch the magnetic field lines and induce a secondary magnetic field i n 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 i n the core and the mantle at the C M B will stretch the magnetic field lines and set up a restoring magnetic force that opposes 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 i n the core, we need a dynamical description of the fluid core. W h i l e 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 between the cylindrical shells. This model is based on an electromagnetic coupling between cylinders and is presented as the Torsional Oscillation ( T O ) theory. T h e 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 will 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 i n 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. T h e magnetic field provides a restoring force and the cylindrical shells will undergo oscillations. T h e 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 w i t h the mantle is accomplished through electromagnetic coupling associated with the u>-effect. Braginsky [1970] provided a quantitative assessment of this coupling, described i n 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 will not be strong enough to break the rigid nature of the fluid motions. In other words, the deformation of the magnetic field i n the fluid does not alter the geostrophic force balance near the ends of the fluid columns. T h e 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]). T h e Lorentz force will then be small and the Proudman-Taylor theorem is valid. T h e 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 ( I C B ) can be substantially larger. Recent dynamo simulations [Glatzmaier and Roberts 1996] suggest fields that can be as large as 100 G. T h e electromagnetic coupling of the cylindrical shells that terminates on the surface of the inner core will 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 i n 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. T h e angular momentum exchange between the tangent cylinder and the mantle is then fully assessed through the electromagnetic coupling by the magnetic stresses at the CMB.  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 m i n i m u m gravitational potential energy between an heterogeneous mantle and the inner core. Misalignments between the mantle and  the inner core will increase the gravitational potential energy and will produce a  restoring force that tends to realign them. This coupling proves to be very efficient and the realignment will 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 i n 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 i n two distinct regions above and below the inner core. These cylinders will 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. A s we have already pointed out, the magnetic field at the I C B is potentially very strong. T h e coupling between the inner core and the fluid will 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 because the large magnetic field at the I C B will generate a large Lorentz force that can no  Chapter 1: Introduction  8  longer be neglected i n the force balance i n the fluid. This will invalidate the P r o u d m a n Taylor constraint and disrupt the rigid character of the fluid motions. T h e coupling model developed by Braginsky [1970] for the C M B , where the rigid rotations were a priori assumed, 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 behavior of the fluid as a result of the electromagnetic coupling w i t h moving conducting solid boundaries. In this model, oscillations i n 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 i n response to this forcing.  The  Lorentz force is kept i n the force balance and the deformation of the magnetic field at the boundaries will 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 w i t h 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 w i t h 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 i n the fluid arise. T h e deformations i n the fluid cylinders are strongest at the boundaries and this raises concern on another issue. T h e calculation of angular m o m e n t u m in the core is based upon the assumption that the zonal velocities at the top of the core are equivalent to the velocities i n the middle of the fluid columns. Large velocity gradients i n  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 instance, 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 oscillation of the L O D 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 i n the dynamo simulation models. T h e 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. T h e 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. T h e equations that describe the evolution of the velocity i n the fluid and the evolution of the magnetic field i n the fluid as well as i n the solid are developed. Simplifications of these equations are achieved by representing the fields i n terms of poloidal and toroidal potentials. T h e final system of coupled equations i n 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 i n the fluid to the solutions i n the solid are developed. In the t h i r d 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 w i t h the electromagnetic coupling.  Among  other things one gains familiarity with the concepts of boundary layers that form i n the fluid as a result of the adjustment of the fluid to the oscillating boundary. T h e principles of fluid transport, electrical current loop and induced magnetic field i n the boundary layer are examined i n detail. In the fourth chapter the second boundary is reinstated. We obtain solutions i n the intended geometry; a fluid cylinder trapped by two conducting solid layers. T h e electromagnetic coupling between the fluid and the solid occur at each boundary. T h e physical  Chapter 1: Introduction  11  insight gained i n 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. T h e presence of these boundary layers is essential for the transfer of angular momentum from the solids to the whole column of fluid. T w o distinct regimes of solution i n the fluid are identified i n the analysis; a diffusive regime and a wave regime. R i g i d rotations persist i n the diffusive regime, where the boundary layers are small and the amplitude of the fluid and magnetic disturbances i n the layers are also small. T h e assumption of rigid rotation breaks down i n the wave regime, where waves travel through the whole column. T h e 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 i n the momentum balance will propel the solution into a wave regime. The fifth chapter assesses the transfer of angular momentum to the fluid by the oscillating boundaries.  It is possible to compare m y results w i t h the angular m o m e n t u m  transferred when rigid rotations are assumed i n the fluid. It is shown how m y results disagree w i t h rigid rotations for a range of magnetic field strength and a range of frequencies of oscillation. The sixth and last chapter introduces a t h i r d forcing term on the fluid; an electromagnetic torque that results from relative motion between adjacent cylindrical shells (e.g. the restoring force which is responsible for torsional oscillations). T h e theory of T O assumes 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 will 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 w i t h the solid boundaries that relies on rigid rotations. This assumption can be verified by  12  Chapter 1: Introduction  incorporating i n m y model a torque concentrated at a particular height i n 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 i n assuming rigid rotations i n the fluid for the purpose of assessing the electromagnetic coupling at the I C B . In the  final  chapter, we will 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 i n the fluid to generate waves of sufficient amplitude. Hence, a rigid rotation model for the flow is sufficient i n 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 i n the m o m e n t u m 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 i n the direction of rotation.  A x i s y m m e t r i c motions are  characterized by relative rotations of fluid cylinders aligned with the rotation axis. W e 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 will 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 will 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. T h e axisymmetry of the system would be preserved i n the curved geometry. The axisymmetrical shear i n 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 i n the equatorial regions of the I C B and the C M B , where the real surfaces are nearly aligned with the rotation axis. T h e vertical extent of the solid above and below the fluid is assumed to be infinite.  Chapter 2: Formulation of the problem  14  E a c h solid "plate" can be considered either as a conductor w i t h a uniformly distributed conductivity or as an insulator with a thin finite conducting layer adjacent to the fluid. T h e whole geometry is i n rotation at a frequency (normal to the plates).  aligned w i t h the axis of the fluid cylinder  T h e rotation rate of the conducting plates fluctuates about the  average value f2 w i t h a frequency ui and their motion will perturb the fluid as a result of the magnetic coupling. We assume that a constant magnetic field B = B z, z  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. T h e other components of the field will alter the momentum balance i n the fluid but not the coupling mechanism. T h e y can be neglected altogether, as results of recent dynamo simulations [Kuang & B l o x h a m 1997] suggests that the poloidal magnetic field at the I C B is mainly aligned w i t h the rotation axis.  z,Cl,B Solid  «*  Fluid  z=0 .Solid  Figure 2.1: T h e 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). T h e fluid is i n a hydrostatic initial state. Perturbations from this basic state are governed by the NavierStokes equation, which gives the momentum balance and describes the evolution of the  Chapter 2: Formulation of the problem  MAN  1  15  MANTLE  LI."  FLUID  FLUID  MANTLb  INNER C O R E  A  B  Figure 2.2: Identification of the bounding solids depending on the location of the cylinder. A : F l u i d cylinders outside the tangent cylinder. B : F l u i d cylinders inside the tangent cylinder.  velocity field i n the fluid [Gubbins and Roberts 1987], <9v — + v V v + 2fixv = -  dt  / P\ V  -  \p)  1 + — ( V x B ) x B + i/V v ,  (2.1)  2  pn  where v is the velocity field, B is the magnetic field, 0 is the frequency of rotation of the E a r t h , 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. W e neglect the effect of viscosity {y = 0) i n the momentum balance as the E k m a n number (importance of the viscous forces) i n 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 i n density and thus no buoyancy force. T h e 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.  T h e velocity field v i n equations (2.1) and (2.2) represents the perturbation from a rotating initial state. The perturbation of the magnetic field from the background poloidal  Chapter 2: Formulation of the problem  16  B field is denoted by b and the total magnetic field will be B = B z-\-h. T h e magnetic field z  z  induced i n the azimuthal direction by the a>-effect can be large, but the axial component of the field will be significantly smaller than B . We assume that both v and b are small z  so that non-linear terms i n the perturbations can be neglected i n both the m o m e n t u m 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 i 2  (2.3)  —b = V x (v x B z) - V x ( i / V x b ) ,  (2.4)  ot  p/j,  d z  where 7r = P/p. Periodic motions of the solid boundaries produce an axisymmetric torque on the fluid, which causes axisymmetric perturbations v and b with an equivalent period. The geometry of the system suggests the use of cylindrical coordinates (£, (p, z) a n d a convenient way to express the perturbation fields is to separate the azimuthal part (toroidal part) and poloidal part using scalar potentials,  v  =[4  + Vx(^)]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 divergencefree 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_ s ds  d -P dt  t  d , - B—ip dz  n  9  (did d > — -£—sP + —P \ds s ds dz > 2  n  n  V  2  „ d ,  n  ( did  We thus have equations of the form -§- [M] z  d  2  n  = 0  >  = o  (2.6)  (2.7)  = 0 and \f- s [M] = 0. It follows that s  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:  (2.8)  -T dt S dt -Ai, dt  - B—u dz  - AT  9  V  B S  2S1  £  dz  (2-9)  = 0  T  pfi dz  =  + 20—u —AP dz pp, dz  where:  A(-)  did., ds s ds  d  2  (2.10)  0  .,  = 0,  (2.11)  Chapter 2: Formulation of the problem  18  If we assume that the geometry extends to infinity i n the radial direction, we can simplify the problem by using a von K a r m a n 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 i n (2.5) yields  1  b  d  z  *  '  (2.12)  = [ - 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. W e note that only the axial component is independent of s. T h e toroidal and radial components increase w i t h 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 will also be unbounded and will induce infinite velocities i n 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. T h e von K a r m a n similarity solution also represents a good approximation of the physical behavior of the system because the departures from a linear gradient i n 5 between the variables of adjacent fluid shells i n the radial direction is expected to be small when the cylinders are threaded w i t h 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 i n equations (2.8) to (2.11) occurs i n the operator A . B y assuming that each scalar has the form  ib(z,s) = stp(z), we  eliminate the s derivatives i n A , leaving only the second derivative w i t h 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  —T —u - riT -~ B B-u-v—f Ot dz 'd  ^0  z  £ i  dt  _  oz  (2.13)  -V^P=0  (2.14)  - ^ | - f = 0  pfi oz  +an*.-5.^=0 ai oz  oz  2  (2.15)  (2.i6)  pp, oz  3  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:  Tt --i'& -=° p  <' >  p  2  =°-  (  2  17  -  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  2.2  20  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 i n z with fixed coefficient. The fluid variables accept solutions of the form  Y = Ye - , iXz  (2.19)  iut  0  where Y is a generic variable and Y its associated amplitude. A solution of this k i n d w i l l 0  generate oscillatory solutions i n both time and distance. The A's are the eigenvalues of the system which, as we will see, have both real and imaginary parts.  Thus the eigenvalues  describe the wavelength of the spatial oscillations and their attenuation or growth. The solution i n the fluid is obtained b y solving the four coupled equations i n (2.13) to (2.16). Substituting the known time and spatial exponential dependency of (2.19) for every variable i n these equations yields a set of algebraic conditions:  -icvP - i\Bj 0  + AV„ = 0  (2.20)  + \ r,f = 0  (2.21)  Q  -iwf - i\B u 0  z  2  0  0  B -ium - U20V> - i\—T = 0  , (2.22)  -iu\ i>  (2.23)  0  o  2  0  - i\2Qu  pp  0  - a ^P 3  0  Q  =0,  Chapter 2: Formulation of the problem  21  where the common time and spatial dependence has been eliminated. T h e above system can be written i n m a t r i x form as: 7 7 A — iuj  0  0  r/X — iu  2  0  z  0  T  z  -IX2Q,  ^ -iX 3 PI*  Po  -i\B  0  2  0  The  -i\B  0  (2.24)  = 0.  —iw  -iu>X -iX2Q 2  U)  0  algebraic conditions i n (2.24) admit a t r i v i a l solution for the amplitudes of the  perturbation, but this is of no interest.  Non-zero solutions can be obtained only i f the  m a t r i x 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. T h e final solution will then be the sum of all the eigensolutions, which we describe as the normal modes of the system. W e note that (2.24) is not a conventional eigenvalue problem, since the m a t r i x is not of the standard form A — XL  For our system, the eigenvalues are not limited to the d i a g o n a l a n d bear  higher powers i n some occasions. Equating the determinant to zero gives a cubic equation i n A : 2  A  7/ (2Q) + 2  2  -ir)u> +  B  2 \ 2N  + A (-2^-oj 4  +A  2  - i2ur)(2n)  2  2  (-u> (2Q) +u> ) = 0 , 2  2  4  + i2o> ?7 3  (2.25)  from which the A's (the eigenvalues) can be found. It is interesting to note that this system is of order 6. A d d i n g 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 will be zero. T h e remaining eigenvalues can be found by numerically solving a quadratic equation i n A . 2  Chapter 2: Formulation of the problem  22  The eigenvalues include both real and imaginary parts, which indicates that the solutions are oscillatory and dissipative. T h e magnetic diffusivity is the sole source of dissipation i n the system and when it vanishes, the fluid becomes a perfect conductor and the waves do not suffer any ohmic dissipation. Setting 77 = 0 i n the dispersion relation i n (2.25) removes all the imaginary terms and reduces the dispersion relation to  A (j^j  + A ( - 2 ^  6  4  2  )  + A (-u, (2ft) .+V) = 0 , 2  2  2  (2.26)  which gives purely real solutions for A. According to the spatial dependence assumed i n (2.19) we obtain oscillatory solutions which are undamped. This last equation is consistent w i t h the dispersion relation for ideal M C waves [Gubbins and Roberts 1987], so called because of the basic pressure-Magnetic-Coriolis force balance i n effect. We note that these waves can be viewed as Alfven waves (Magnetic-pressure force balance) modified by the rotation of the E a r t h . W h e n dissipative effects are included the nature of the eigenvalues will include 2 growing and 2 decaying waves. T h e growing solutions present a problem because the dissipative effects should attenuate the disturbances, not intensify them. B u t 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  YC Y e - °- \ iXn{z z ) iu  =  j  n  (2.27)  n  n=l  where z is the position of one of the plates. Y represents the eigenvector associated w i t h Q  n  the eigenvalue A and C n  n  is an integration constant. T h e 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 w i l l be developed in chapter 3 and chapter 4, respectively. This mode will generate spatial solutions that are either constant or linearly dependent on z. The remaining modes will be characterized by attenuated oscillations i n z.  Solutions of the diffusion equations i n the solid (2.17 and 2.18) are also of the form:  Y. = Y„. ^ ' " ^ >  (2-28)  but unlike the fluid equations, the equations i n the solid are uncoupled and we can directly solve for the common eigenvalues X for both the poloidal and toroidal perturbations: s  A, = ± ( i + 0  The inverse of \  s  (2.29)  2^  gives the wavelength of the oscillation i n 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 i n the solid is the sum of a growing  oscillation and a decaying one:  '  p /  J,X$(z-z )-iut 0  _|_  J.\  a  (z-z )-iwt  (2.30)  0  f: Since the fluid-solid interface is located at z = z , the 4 constants ( P , P~, T+ and +  0  s  r~) are i n 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 i n 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 C  n  i n (2.27) are then the only remaining unknowns i n the  fluid, while the amplitude constants i n (2.30) are the only unknowns i n the solid. A l l these  24  Chapter 2: Formulation of the problem  constants are constrained by the boundary conditions, which are examined i n the next section. These boundary conditions link the solution i n the solid plates to the solution i n the fluid.  2.3  Boundary conditions  Solutions for the fluid and solid systems can be found i n terms of a number of arbitrary constants. fluid-solid  To specify these constants, we have to impose boundary conditions at the interface.  T h e component of the magnetic field perpendicular to the boundary, the axial magnetic field,  must be continuous across the interface.  T h e 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 until after we have dealt w i t h 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 i n 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 i n the fluid and the magnetic field perturbations i n 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 i n the interior of the fluid. T h e adjustment of the fluid velocity to the no-slip condition forms a viscous boundary layer i n the fluid, while the adjustment of the ambient magnetic field produces a magnetic boundary layer i n 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 will 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 i n 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.  T h e basic balance i n 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. T h e kinematic viscosity is poorly known, but it is believed to be roughly 1 0  - 6  m s 2  _ 1  [Gubbins and Roberts 1987]. For phenomenon  occurring during a time scale on the order of days ( f t ) , the viscous layer w i l l then be at - 1  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 - ft , the magnetic 4  -1  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. T h e 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. T h e 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. T h e discontinuity i n the tangential velocity does not affect the continuity conditions on P and  T. T h e continuity of the magnetic field scalars has to be preserved i n time. Hence, the time-evolution of these scalars should also be continuous. The evolution i n time of both magnetic field scalars is given by the induction equations (2.13) and (2.14). T h e integration of these two equations across the fluid-solid interface, and thus across the velocity discontinuity, give two more boundary conditions, which relate the j u m p i n 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 j u m p i n the gradient of the magnetic field i n relation w i t h the j u m p i n velocity, which are obtained from the poloidal and toroidal part of the induction equation:  1> =  0  (2.31)  P = Ps  (2.32)  T = T  (2.33)  S  Chapter 2: Formulation < the problem  27  r  d -T ~rj-T oz  V  s  d -•" = 0 oz  (2.35)  In the fifth boundary condition, the discontinuity i n the toroidal velocity across the boundary introduces the velocity of the plate, u . This plate velocity is a parameter of the 3  system and may be viewed as the forcing term. T h e amplitude of the fluid perturbation w i 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 i n the solids. This means there can be no force on the solids and hence, no reaction force on the  fluid.  T h e 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 will carry the magnetic field. Conversely, the magnetic field will slip easily through a poor conductor. W h e n the conductivity of the plate drops to zero, the B field is undisturbed by the motion of the z  plate. Since we have neglected the effects of viscosity, the fluid will not be perturbed. The model of the conductive plates that extend to infinity requires both P and T to s  3  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 w i t h 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. T h e 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 i n the conductive layer, but the continuity of the magnetic field requires the solution to match the scalar potentials of the field i n the insulator at the conductive layer-insulator interface. A l l electrical currents  28  Chapter 2: Formulation of the problem  must vanish i n the insulator, which i n our notation translates into the conditions: Tj = 0 and -Q^- = 0. T h e scalar P  T  could be represented by a linear term i n z plus a constant,  but the requirement that the solution remains finite at infinity requires the perturbation of the axial magnetic field i n the insulator to be simply Pj = C, where C is a constant. T h i s spatially constant field will oscillate i n time with the imposed periodicity. T h e continuity of the magnetic field at the boundary between the conductive layer and the insulator imposes three conditions: P = P r , ^ s  = ^  t  and T = T . K n o w i n g the solution i n the insulator, 5  T  these conditions can also be written: P . = C, ^ p = 0 and T „ = 0. These two models for 1  3  f (jz  S  the solid are interchangeable i n the calculations and the choice of one over the other can affect the solution i n the  fluid.  29  Chapter 3 Single-plate problem 3.1  Introduction  The first case we will 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 i n z. This simple case is useful for developing insights into the electromagnetic coupling between a fluid and a moving boundary.  A s we have already  hinted by physical arguments, and by the nature of the solution, the perturbation i n the fluid is periodic i n time and i n space. The attenuation of the waves w i t h distance from the boundary defines a region called a boundary layer, where the flow adjusts to a boundary perturbation. The presence of boundary layers i n 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 E k m a n layer is a result of a viscous friction between the fluid and the boundary when the whole system is i n rotation (the Coriolis force is important i n the force balance) [Pedlovsky 1987]. This boundary layer has been extensively studied and used i n oceanography. If the fluid and the boundary are subject to electromagnetic coupling i n addition to the viscous coupling, then the fluid velocity will be influenced by an additional adjustment of the magnetic field. T h e imposed magnetic field is dragged through the fluid by perturbations i n the motion of the plate. T h e magnetic forces alter the force balance i n the boundary layer, which, i n the absence of rotation, is often called the H a r t m a n n layer [Gubbins and Roberts 1987]. W h e n the system is i n rotation, this layer can be regarded as an E k m a n layer w i t h the influence of an imposed  Chapter 3: Single-plate problem  30  magnetic field or a H a r t m a n n layer to which we have added the effect of rotation. This k i n d of layer is often called an E k m a n - H a r t m a n n boundary layer [Gubbins and Roberts 1987]. Loper [1970] studied the hydromagnetic boundary layer induced by a rotating conducting plate.  His calculation shows the presence of an E k m a n - H a r t m a n n layer i n 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 region, hence, beyond the E k m a n - H a r t m a n n 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 E k m a n - H a r t m a n n layer by making suitable adjustments to the boundary conditions, as we argued i n the previous chapter. T h e coupling between the fluid and the solid is then solely accomplished by magnetic interaction and the ensuing boundary layer region i n 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 i n the solid. T h e solution i n the fluid is given by (2.27). W i t h o u t the second plate bounding the fluid vertically, the solutions that grow endlessly towards infinity are excluded. Hence, the two eigensolutions w i t h eigenvalues Im(\)  < 0 are eliminated from the final solution. T h e expansion of (2.27)  Chapter 3: Single-plate problem  31  is then the sum of three normal modes: •  p  Pi  f  e~  iwt  AXr.z—iuit e  + C  2  i\~z—iwt e  2  3  (3.1)  V>3  u where C ,C 1  2  and C  u  2  3  are the unknown integration constants.  T h e modes w i t h the non-  zero eigenvalues will give rise to oscillatory solutions that decay w i t h z.  T h e physical  significance of the mode A = 0 can be inferred by looking at the structure of the eigenvector. Substituting A = 0 i n (2.24) gives  0  -iu>  0 0  0  0  — IU!  0  0  0  0  Pi (3.2)  which yields:  A  0 0  +i  1  «i  0  (3.3)  for the first eigenvector i n (3.1). F r o m the representation of the velocity i n (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 will be shown that this axial velocity is connected to a radial transport i n the vicinity of the plates. Because the fluid is incompressible, a sink of fluid calls for a source elsewhere i n 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 i n the boundary layer. This phenomenon is often referred to as pumping. T h e amplitude of this pumping will 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 i n the solid to impose boundary conditions on the magnetic field and solve for the integration constants C i n (3.1). If we assume a uniform conducting n  infinite plate, then we have to eliminate the eigenvalue \  3  w i t h i m ( A ) > 0 i n (2.29) so 3  that solution i n the solid decays towards z = —oo. T h e solution i n the solid (2.30) is then: -  P.  J.\,z—iut  (3.4)  where  A, = - d  +  i ) ^ .  (3.5)  We have a total of 5 unknowns i n the solutions (3.1) and (3.4): there are 3 unknown integration constants (C , 1  C, 2  C ) i n the fluid; and the amplitude P~ and T~ of the 3  perturbed magnetic field i n the solid at the boundary.  T h e boundary conditions listed  i n (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 i n m a t r i x form  Chapter 3: Single-plate problem  33  as  V>2  Pi  P*  fl  0  0  n  -1  0  0  0  -1  0  v\Pi -7i  0 -72  -7s  A"  0  0  where 7 = z?7A T + 75 'u . Presented i n this form, it is clear that the prescribed oscillation n  n  of the plate (u )  n  2  n  is driving the whole system.  s  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 w i t h the periodic time dependence: e~  .  lu>t  Several comments can be made on the boundary conditions. We have explicitly eliminated the exponentially growing eigensolution i n the fluid. We could have accomplished the same task by requiring the solution to be defined at infinity. These regularity conditions 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 t h i r d condition is required. The additional condition is imposed on the velocity at infinity. If the magnetic field is undisturbed, then the main field B z z  cannot be disturbed by any  velocity. Consequently, the radial velocity must vanish: | j = 0 at z = oo, although an axial velocity, parallel to B z, z  is still permitted. This last condition could also be written  as u = 0, using the relation i n equation (2.22) i n the absence of a toroidal magnetic perturbation T. These three conditions force the amplitude factor of the two exponentially growing solutions to equal zero.  34  Chapter 3: Single-plate problem A complication is introduced if one adopts a conductive layer i n 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: T = 0, -gf = 0 and P t  a  equal to a constant  value (say K) i n the insulator. The three additional unknowns would have been P + , T  + s  and K. T h e dimension of the matrix i n (3.6) would then be 8 x 8.  3.3  Results and discussion  The amplitude of the solution i n the fluid is determined by the amplitude of the plate motion, while the form of the solution depends on the parameter values that appear i n the m a t r i x i n (3.6). Since we are more concerned w i t h 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 w i t h a 1 degree (<f> = 7r/180 radians) amplitude. T h e motion of the plate is:  d<[>  u=---= 8  dt  OJ  7T  180  _.  e  (3.7)  t  V  ;  and the m a x i m u m velocity of the plate occurs at t = 0. For an initial example of the solution, it is convenient to use a set of parameters which are not likely to break the Proudman-Taylor constraint. T h e physical processes responsible for the coupling are more easily identified i n 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 initial solution will be calculated with B  z  = 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 E a r t h [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 T O ) . The period of oscillation has been chosen to correspond to the period of L O D 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 I C B 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 10 kgrn' 4-7T x. 1 0 kg m A~ s~ ZA x 10 Sm' 3.4 x l 0 5 m " 7.27 x l O - ^ 5G O/10 = 7.27 x I O " * WTT/180 = 1.26 x I O * " 4  3  - 7  2  s  1  B  1  5  B  Z  u>  2  1  4  9  1  - 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 L O D variations is u/u ; the toroidal velocity s  Chapter 3: Single-plate problem  36  reflects the angular momentum distribution i n the fluid, which is the quantity we want to investigate to deepen our understanding of momentum transfer i n the different regions of the Earth's interior. T h e amplitude of u varies periodically w i t h z, as the disturbance due to past plate movements propagates away from the boundary. T h e force responsible for the coupling between the fluid and the plate is a magnetic stress at the fluid-solid interface. T h e propagation of the disturbance away from its source at the boundary is limited to a finite region by diffusive losses. This boundary layer region i n the fluid can also be referred to as the magnetic diffusion region, to follow the terminology used by Loper [1970]. T h e 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. T h e decaying-oscillating behavior i n this boundary layer is attributable to the second and t h i r d mode of the solution i n (3.1). T h e 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 E a r t h . A l l 4 scalars i n the fluid have similar curves, which is expected given the form of the solution i n (3.1). A boundary layer is also formed i n the solid but it involves only the magnetic field scalars. T h e amplitude of the magnetic field scalars i n b o t h the  fluid  and the solid are of the same magnitude, as required by the boundary conditions. T h e y also both follow a solution governed by diffusion; i n the solid, because the magnetic field is represented by a diffusion equation; and i n the fluid, because we are i n the diffusive regime. T h e first mode i n (3.1), the constant vertical velocity, only affects the solution of ip. We can see that this scalar does not decay to zero w i t h distance but approaches a constant velocity. Since the eigenvalue of this first mode is zero there is no oscillating or decaying  37  Chapter 3: Single-plate problem behavior. Instead, the velocity is maintained throughout the whole space i n z. T h e behavior of ip can also be related to the solution of u w i t h simple physical ar-  guments. In the boundary layer, the oscillation of ip represents vertical fluid movement. A s equation (2.12) indicates, the derivative of ip w i t h respect to z represents the radial velocity, so we can deduce from the spatial variation i n ip that the radial flow will also vary w i t h z i n 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 i n the vertical direction. R a d i a l motion also affects the toroidal displacement of the fluid by conservation of angular momentum. For example, when the radial pumping brings i n 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, because of magnetic forces which can transfer angular momentum (see equation (2.22)). T h e 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 w i t h the oscillating behavior of u through angular momentum conservation and fluid circulation. We now have the physical insight needed to explain the first mode. T h e fluid adjustment to the electromagnetic coupling with the moving plate produces a net radial transport of fluid i n (or out through) the boundary layer. This is compensated by a vertical p u m p i n g of equivalent mass out of (or into) the layer. B u t there can be no vertical transport  at  the boundary w i t h 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 i n the boundary layer. We recall that the solutions presented are snap shots i n time. This pumping oscillates w i t h time and half a period later, it will be completely reversed as the radial pumping i n the boundary layer will also be reversed. T h e circulation of the conducting fluid i n 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. T h e toroidal velocity u will stretch the B field i n that same direction, z  setting up a magnetic field that will produce a force that opposes the deformation. T h i s secondary magnetic field is toroidal (T) and consistently w i t h the oscillating u, it presents an oscillating-decaying behavior. It is also common to represent this secondary field i n terms of electric current, which mathematically is expressed as the curl of the magnetic vertical current aligned w i t h B  z  field.  Associated w i t h T is a  (figure 3.1 A ) . This current points upwards (downwards)  for positive (negative) T, and follows a similar oscillating-decaying behavior. T h e variation of T along z demands the presence of a radial current (|jj, figure 3.1 B) which has a similar oscillating behavior i n z. In the same way that the vertical and radial velocities are connected w i t h one another to satisfy the fluid mass conservation, the vertical and radial currents are also related. T h e 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 i n a "disc" volume will be compensated by outward vertical currents and vice versa. T h e magnetic force results from the cross product between the induced current and the magnetic field, which is approximated by the background field B  z  i n the linearized  case. T h e radial current (figure 3.1 B) creates the expected toroidal force but the vertical current (figure 3.1 A ) is parallel to B , nullifying the cross product. Hence, only the radial z  Chapter 3: Single-plate problem  39  component of this poloidal current will contribute to the induced magnetic force. The  presence of P can be understood in a similar way. T h e radial motion i n the  boundary layer deforms the B field which sets up a poloidal secondary magnetic field that z  induces a counteracting radial force. The curve of P follows the behavior imposed by ip. A g a i n , we can look at it also from the point of view of secondary electric currents. T h e induced poloidal magnetic field is associated with a toroidal current (-3-3-) figure 3.1 C ) . This toroidal current satisfies charge conservation. T h e cross product of this current w i t h B  z  will indeed give a force i n the radial direction which opposes the deformation of the  main field by the radial flow.  c  Figure 3.1: T h e secondary magnetic field is grey shaded and the associated electric current is black. A : T h e 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 : T h e  vertical gradient of the radial field ( § j r ) is associated with a toroidal current.  In summary, fluid motions i n the boundary layer induce currents which i n t u r n alter fluid motions. T h e 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. T h e vertical current generates no force and no current is able to generate a vertical force. However a vertical flow develops i n 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. T h e shear i n the toroidal velocity near the plate deforms the background axial magnetic field, creating a secondary toroidal magnetic field (T). This field will 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 i n the boundary layer i n the fluid and an equal but opposite net radial current i n the diffusion boundary layer of the solid i n order to conserve charge. T h e net radial current i n the fluid interacts with B  z  accelerates the fluid.  to produce a toroidal force that  Angular momentum is then transferred from the solid to the fluid  by the exchange of charges between them. W i t h o u t viscous effects, electromagnetic i n 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 i n the fluid and the toroidal Lorentz force also vanishes. This result is i n 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 H a r t m a n n current.  Chapter 3: Single-plate problem  41  T h e force due to the current across the fluid-solid interface is often described i n terms of the magnetic stress imposed on the fluid by the moving boundary. T h e magnetic stress is proportional to the product of the axial magnetic field (B )  and the toroidal deformation of  z  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 i n the boundary layer. T h e radial transport of fluid i n 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 i n the boundary layer is subject to the Coriolis force, which tends to deflect the motion i n 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 i n the  A  =  fluid,  ^  %•  W i t h a weak magnetic field (B  z  = 5G), this dimensionless number is small.  T h e force  balance i n 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 B  z  by the velocity, but this source is negligible compared w i t h the diffusion.  Similarities between the purely diffusive solution i n the solid and the solution i n the  fluid  indicates that the fluid motions have little influence on the magnetic field. A l t e r i n g the frequency u> will alter the inertia of the  fluid.  A l t h o u g h , 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) will also be unaffected.  Hence, the solution remains i n a diffusive regime and the shape  of the curves for every variable will be identical. In figure 3.3 and 3.4 we present results obtained respectively w i t h m = O / 1 0 and ui = 2  fi/10 . 6  The curves differ only i n the scale  of the disturbance i n z. The magnetic skin depth (y2ri/tjj)  depends on the frequency, and  slower plate movement will permit a deeper propagation of the perturbation i n b o t h 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 will decrease. Changes i n inertia can be seen by looking at the amplitude of the velocity scalars. The absolute amplitude of the toroidal velocity i n the fluid will be different, but its value relative to the plate motion remains unchanged. T h e strength of the magnetic coupling that allows the plate motion to drag the fluid i n the toroidal direction depends on the strength of B  z  and not on the frequency. T h e absolute amplitude of ip varies w i t h the changes i n  the absolute amplitude of u. For a higher (lower) frequency, the pumping has to increase (decrease) to accommodate the increase (decrease) i n the toroidal velocity. However, the change i n amplitude of ip is not directly proportional to the variation i n u>. Conservation of angular momentum, which relates ip to u, does involve u>. Consequently, the frequency will alter the ratio between the toroidal velocity and the radial velocity. The changes i n inertia will also affect the secondary magnetic field. For a higher frequency, 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. T h e increase i n the radial velocity associated w i t h a faster plate motion necessitates a stronger radial magnetic force. However, the absolute amplitude of the secondary poloidal field P stays constant for variations i n frequency. A n increase i n frequency  Chapter 3: Single-plate problem  43  narrows the boundary layer and the increase i n the radial magnetic force is accomplished solely through this change i n vertical length scale.  A lower frequency has exactly the  opposite effect on both the toroidal and poloidal secondary magnetic field. For phenomenon w i t h periods similar to the rotation of the E a r t h , inertial effects will become much more important. W h e n the force balance is between the Coriolis force and the inertia, inertial waves will be generated i n the system. O u r 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 w i t h small temporal Rossby number. The relative importance of the Lorentz force versus the Coriolis force will be changed if B  z  is altered. For B  smaller than 5 G , the momentum equation continues to balance  z  Coriolis force and pressure gradients, while the magnetic solution will still be i n the diffusion regime (figure 3.5 w i t h B  z  = 0.5 G). The coupling between the plate and the fluid will be  less efficient and the amplitude of each variable will decrease. B u t the shape of the curves will not be altered because the magnetic skin depth which defines the characteristic length of the disturbance remains unchanged. When B  z  is stronger, the force balance is altered because the Lorentz force becomes  more important. T h e 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. T h e presence of a restoring force i n the momentum balance alters the fluid motions, which induce a magnetic field perturbation and overcome diffusive losses.  T h e time scale for the attenuation of the  disturbance is now several periods of oscillation of the plate (figure 3.6 w i t h B  z  = 50 G).  Thus, an M C wave can be maintained for a longer vertical length scale. T h e solution is  44  Chapter 3: Single-plate problem therefore i n a wave regime. The velocity of these waves will increase w i t h a stronger  B\ z  the perturbation propagates deeper i n the fluid because the waves are moving faster and because the wavelength is longer (so diffusive losses are reduced). T h e magnetic field scalars i n the solid continue to obey a diffusion equation.  The  amplitude of P and T , however, will increase i n step with the changes i n the amplitude s  of P and T.  s  Substituting the model of the uniformly conducting infinite plate w i t h the  model of an insulating infinite plate under a conductive layer will not alter the solution i f the thickness of that layer is larger than the magnetic skin depth i n the solid. However, i f the thickness of the conductive layer is smaller than the skin depth, the coupling is reduced. T h e effect is similar to a simple decrease i n 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 B , z  as long as the temporal Rossby number is small.  W h e n Cl, p and <7y are given, the Elsasser number (3.8) depends only on B . z  B  z  T h e value of  at which the Elsasser number roughly equals unity defines the point where the Lorentz  forces will be comparable to Coriolis force i n 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 limit, 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 i n 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 i n these toroidal motions. This is inconsistent w i t h the idea of rigid toroidal motions. T h e potential problems with the T O theory can thus be expected when we are i n 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. W h a t is also clear from this study is that the transfer of angular m o m e n t u m to the fluid from a single plate is limited to a boundary layer. Hence, even i f the rigid rotations are i n 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 w i t h a second plate that bounds the fluid. T h e two plate model is developed i n the next chapter.  Chapter 3: Single-plate problem  46  U / U<  0  2  (m/s)  x 10"  8  200  5  (G)  x 10"  0  (G/m)  5 x 1  o"  7  Figure 3.2: Solution for the scalar potentials i n the fluid and the solid, obtained w i t h the set of parameters listed i n table 3.1. The vertical axis of every graph represents the vertical distance z. T h e position of the fluid-solid interface is at z = 0. T h e solution presented is at t = 0.  Chapter 3: Single-plate problem  47  Figure 3.3: Solution obtained with the set of parameters listed i n table 3.1, but substituting u> by ft/10 . The vertical axis of every graph represents the vertical distance z. T h e position of the fluid-solid interface is at z = 0. The solution presented is at t = 0. 2  Chapter 3: Single-plate problem  48  u / u<  -20  -15  -10  -5  0  5  (G/m) xio"  8  ^-1000  -2000  -15  -10  5  0  (G/m) xio'  5 8  Figure 3.4: Solution obtained w i t h the set of parameters listed i n table 3.1, but substituting oj by fi/10 . The vertical axis of every graph represents the vertical distance z. The position of the fluid-solid interface is at z = 0. T h e solution presented is at t = 0. 6  Chapter 3: Single-plate problem  49  Figure 3.5: Solution obtained with the set of parameters listed in table 3.1, but substituting B 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. z  Chapter 3: Single-plate problem  2000  50  u / u<  2000  -0.5  0  0.5  1  (G/m) x 10"  5  -2000 -0.01  0.03  -2000  0 (G/m)  5 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.  51  Chapter 4 Two-plate problem 4.1  Introduction  One of the dominant features that comes out of the single-plate analysis i n the previous chapter is the constant axial pumping outside the boundary layer region. T h e pumping extends to infinity i n z and presents a clear problem for the model of the Earth's interior. The next level of improvement i n the model is to introduce a second oscillating plate that will bound the fluid i n z. This will then confine the vertical pumping to a finite region and give a more sensible model for fluid motions i n 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 will arise on the fluid near the second plate and we can therefore anticipate the formation of a second boundary layer. E a c h boundary layer will 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 i m p l y the existence of a flow through the fluid-solid interfaces. T h e solution in one boundary layer has to be consistent with the solution i n the other boundary layer. Hence, the solution i n this finite column of fluid must encompass this idea of communication between the boundary layers i n 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 i n figure 2.1. We define z = 0 at  Chapter 4: Two-plate problem the  fluid-solid  52  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. T h e plates are oscillating w i t h the same prescribed frequency u>, which imposes the same periodic time dependence on the fluid solution. T h e 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 i n (2.13) to (2.16). A s before, we are looking for exponential solutions i n space w i t h a known timedependence 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 i n (2.20) to (2.23). Hence, the eigenvalues and eigenvectors of the double plate system will 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 i n order to satisfy boundary conditions. A fourth and a fifth independent solution must be added to (3.1). We must also deal w i t h the degenerate eigenvalue at A = 0 i n order to obtain two independent eigenvectors. This will 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. T h e 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 i n the boundary layer. In the doubleplate 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 i n phase), the net axial pumping is then zero, which  Chapter 4: Two-plate problem  53  implies the absence of a net radial transport i n 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 w i t h 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 Y i n (3.3) be this first solution, then the second x  independent solution is given by  (f  + zY ) x  6  e**  ,  (4.1)  where Y is an unknown constant vector. 6  Inserting the second independent solution i n the system of equations gives us the vector  0 (4.2)  0  ^6  u  6  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. T h e addition of the second solution w i t h constant vector Y introduces the possibility of a non-zero P and u outside the boundary 6  regions as well as a linear variation with z i n the vertical pumping (ip) associated w i t h  zY . 1  T h e full solution i n the fluid is expressed as a linear combination of its 6 independent  Chapter 4: Two-plate problem  54  parts as: •  •  p  A  f  T  x  Pi  P  G  e~ + C iwt  e~  6  fi  + zC  i w t  2  f  2  e~ + C iut  2  6  i>i  P  i>i  u  U~  u  6  +c*  T  3  giA z—iujt 3  +c  2  e  iA (z-/i)-iwt 4  I Q  =  * A ( z — h)—iuit  (4.3)  5  4  V>5  where the C ' s are integration constants. We recall that the pair of eigenvalues A and A 2  are equal but opposite, where i\  4  has a negative real part and z A has a positive real part.  2  4  T h e same can be said about the pair A and A . 3  5  T h e 6 integration constants are determined w i t h the boundary conditions, which i n troduce additional constants for the solution i n the solid plates.  T h e magnetic field i n  each plate is modelled by the diffusion equations (2.17) and (2.18). E a c h 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. T h e solution i n 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 (2.29) for the bottom plate and the negative root of \ s  s  for the top plate. If we denote by A and B the bottom and the top plate variables, the solution i n the solid is expressed as: pA  OS  rjhA  s  rjhA OS  pB  i\fz-i*t  e  OS  a  n  d  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). A n 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:  _A  3  A  pA  B  F  A  2  C  A  4  C  B  where the A ' s are matrices. The matrices A and A represent the single plate coupling n  x  4  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 A is y  presented in (3.6), and is repeated here for convenience,  i>2  1>1  Pi  p  3  v\Pi  n  = ir)\ f n  n  0  0  -1  0  0  -1 0  7/A3P3  -ll where j  4>3  -7  + Bu . z  n  2  0  -7s  Matrix A is equivalent to A 4  lt  but is evaluated at the top  Chapter 4: Two-plate problem  56  plate and involves only the scalars that decay away from this boundary:  V<5  V>4  6  p*  h  + hfa  Tp  Pe + hPi f  + hf  t  6  0  0  -1  0  0  -1  T]X P - ivP 1  -7  6  0  -  -7e  4  0  1  Matrices A and A represent the influence of the second plate on the single plate 2  3  problem. They are given by  V> e  4e  -P e  4  P&  T e  4  4  4  4  4  77A P e 4  -7  4  e 4  ipe + h^  5  P + hP 6  v^5 s 5 p  4  4  e  V^Pe - " 7 - P i  0 0  ~le  0  -V?!  1>i  4>2 i  ^3 3  0  0  Pi  P2 t  -P3 3  0  0  Te  ^3 3  0  0  0  0  0  0  e  e  2  e  e  2  rj\ P e  v\Pi  2  2  r]\ P e  2  3  -%4  -7i  h  A  3  3  -%4  where e± = e " . Vectors C ,C , ±iX  0 0  e  e  F  B  A  0 0  1  fe + h^  ~% 5  As =  0 0  and F  B  0  are, respectively, the vector of integration  constants and the forcing vector for the single plate problem at each boundary:  C  A  — [Cl->C i 2  ^ M ]  c = [c ,c ,c ,P ,f } B  B  4  F=  [0,0,0,0,  F=  [0,0,0,0,-B u?]  A  ^ 3 ) Posi  5  6  B  0 s  T  B  -B uf]  T  z  s  T  Chapter 4: Two-plate problem  57  We have seen i n the analysis for a single plate that adopting a conductive layer rather than assuming a uniform conductivity i n 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. T h e 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 will be broken. A s i n the single plate case, it is more convenient to begin the analysis w i t h a set of parameters that reproduce rigid rotations. These solutions are obtained i n a diffusive regime, which implies that the imposed magnetic field is weak. We model a fluid column inside the tangent cylinder, so the b o t t o m and top plate will 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. 10 S m~ 5  l  and o~f = 3.4 x 10 S ' m 3  - 1  W e use af  = 3.4 x  i n our initial example solution, and assume that b o t h  plates are uniformly conducting and extend to infinity (e.g. we solve the m a t r i x system i n equation (4.5)). T h e 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. T h e other parameters are the same as those used to produce the initial solution for the single plate case, and are listed i n table 4.1.  Chapter 4: Two-plate problem  58  1.1 x IO kgmAir x I O kgm A' s~ 3.4 x 10 Sm' 3.4 x 10 Sm' 3.4 x 10? Sm4  p  3  2  - 7  /*  0~  °*  *?  s  1  5  1  2  1  7.27 x I O " * " 2000 km 5  h B  1  5G  z  ft/10  4  = 7.27 x l O - ^ 9  O;TT/180 = 1.26 x I O I O " WTT/180 = 1.26 1  - 1 0  x IO"  * 1 1  1  - 1  s-  1  Table 4.1: Parameters used i n calculation of the electromagnetic coupling between a cylinder 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 i n table 4.1. O n l y 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 i n the fluid.  These boundary layers are magnetic diffusion regions, as i n  the single plate analysis. The wavelength of the oscillation is identical i n 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 i n 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.  T h e magnetic diffusion regions at each boundary  thus have identical thicknesses, regardless of the conductivity of the adjacent solid. Similarly, the depth at which the magnetic perturbation propagates i n the solid depends on the conductivity i n this solid and not on the conductivity i n the adjacent fluid. T h e mag-  Chapter 4: Two-plate problem  59  netic diffusion regions i n 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 b o t h non-zero outside the boundary layers. In addition ip shows a linear variation i n z. These features represent the characteristics of the second independent solution associated w i t h A = 0 (see (4.1) and (4.2)). T h e region between the boundary layers was designated as the current free region by Loper [1971]. W i t h the physical insights gained i n the single plate case, we are i n position to explain physically the behavior of these variables i n this current free region. W e recall that the radial transport i n 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 i n both boundary layers. Hence, at each boundary, there is an exchange of fluid mass between the boundary layer and the current free region. W h e n 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 i n the current free region is required to balance the fluid flow. T h e vertical pumping, ip, varies linearly w i t h z i n the current free region as dictated by (4.1). T h e radial transport, which according to (2.12) is defined as the gradient of ip i n z, is then constant everywhere i n the current free region. A s shown i n the cartoon of figure 4 . 1 A , a poloidal circulation is then set up i n the whole column of fluid. T h e second independent solution associated w i t h A = 0 (4.1) is then responsible for the fluid circulation required to conserve fluid mass.  T h e first independent solution for  A = 0, which represents a constant axial velocity i n z (3.3), fits the p u m p i n g variations i n the current free region to the appropriate values of pumping at each end of this region (at the limits of the magnetic diffusion regions).  60  Chapter 4: Two-plate problem  We have already discussed i n the analysis of the single plate that the radial transport i n the boundary layers is associated with a toroidal acceleration to conserve angular momentum. T h e same concept applies i n the current free region, where we now have a constant radial transport.  Hence, the whole region of fluid between the boundary layers  will 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 i n the current free region.  A  B  Figure 4.1: Toroidal acceleration of the current free region due to fluid circulation. A : T h e magnetic stress at the boundaries produces a radial transport i n the boundary layers (black arrows). A poloidal circulation is required i n the current free zone to conserve mass. B: T h e radial transport i n the current free region produces a toroidal acceleration by conservation of angular momentum.  In the current free region, there is no shear i n the toroidal velocity and hence, no deformation of B . The constant radial transport will either concentrate the B z  z  field lines  together or "move them apart". This modifies the intensity of the axial magnetic field and causes a perturbation to B . z  This explains the presence of the constant P i n 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". T h e currents are confined to the magnetic diffusion regions i n the fluid and the plates.  Chapter 4: Two-plate problem 4.3.2  61  Transfer of angular m o m e n t u m  A t each end of the column, electromagnetic coupling can transfer angular m o m e n t u m from the solid to the fluid. T h e 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. W h e n the net radial current i n one boundary layer is not equal and opposite to the net radial current i n 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 w i t h Taylor's theorem [Taylor 1963], which stipulates that i n 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":  (4.6)  In our axisymmetrical geometry, B = B z so only the radial current will interact w i t h z  this magnetic field to produce a torque. W h e n 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. T h e 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 i n the single plate analysis is a net radial transport of fluid i n or out through the boundary layers i n the fluid. A s explained  62  Chapter 4: Two-plate problem  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 limited to the magnetic diffusion regions i n both the fluid and the solid. B : The net radial current i n the boundary layers i n the fluid produces a torque, which accelerates the fluid toroidally.  in the previous subsection, conservation of mass requires a constant radial transport i n 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 B  z  field is undeformed i n the  current free region, this toroidal acceleration can only be rigid. As the fluid i n the boundary layers adjusts to the electromagnetic coupling w i t h 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 i n 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] i n his analysis of the viscous spin-up of a fluid layer enclosed by two rotating flat plates. T h e spin-up time of the whole fluid was derived from the E k m a n 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 pumping 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 i n the same direction, then there is no need for a radial transport i n the bulk of the  fluid.  A n d thus, no toroidal acceleration i n the current free region. A g a i n , this will happen when the magnetic torques at each end are equal but opposite (e.g. when the plates have the same conductivities and are oscillating w i t h the same amplitude but a phase difference of exactly 180 degrees). T h e angular momentum carried by each boundary layer is equal and opposite, so the net radial transport i n the boundary layers is ehminated. F l u i d mass is conserved without requiring a radial transport i n 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 determined. T h e fluid circulation enables the plates to communicate w i t h each other. Thus the amount of charge exchanged at each boundary will adjust to the total angular m o m e n t u m transferred to the fluid column. It is possible to express the intensity of the axial current at one boundary i n terms of  Chapter 4: Two-plate problem  64  parameters of our system. W h e n the magnetic field is small, the magnetic restoring force is negligible i n 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 diffusion equation ( T ) , the solution of the toroidal part of the diffusion equation i n the solid (T ), s  w i t h the boundary conditions (2.35) and (2.33), we find a relationship for the  amplitude of the axial current at the boundary i n terms of the parameters of our model. This relationship is:  (4.7)  where oc denotes proportionality, and conductivities er and o-j refer to the solid and the s  fluid, respectively. T h e quantity u — u  a  represents the differential rotation between the  fluid and the solid at the interface, with u being the velocity of the rigid rotation i n 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 i m p l y that the magnetic stress is significantly different at each boundary. For instance, i n figures 4.3 and 4.4, the conductivity of the bottom plate is much larger. T h e 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 b o t t o m plate. However, the top plate having a toroidal velocity 10 times smaller than that of the b o t t o m plate, the shear i n velocity is much larger at the top boundary. T h e decreased conductivity of the top plate is compensated by larger j u m p i n 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  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 i n the solid would necessarily decrease the magnetic stress at the boundary. A smaller toroidal velocity of the plate would also have the same effect.  B u t i n the bounded geometry, the magnetic stress at one boundary is not inde-  pendent of the magnetic stress at the other boundary. T h e y both adjust to the angular m o m e n t u m 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 i m i t e d 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. E v e n the amplitude of the oscillations i n the toroidal velocity is a few orders of magnitude smaller than the rigid toroidal motion. T h e thickness of the magnetic diffusion regions can be extended by reducing the period of oscillation of the plates, as we have shown i n the previous chapter.  However, the  amplitude of the oscillations will still be negligible compared w i t h the rigid rotation velocity. R i g i d rotations still represent an accurate description of the flow. W i t h a weak magnetic field, the solution is i n a diffusive regime.  Changes i n 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 i n the  fluid  enables waves to propagate although the velocity of their propagation is slow and they are rapidly attenuated. The Proudman-Taylor constraint is still applicable and the basic  Chapter 4: Two-plate problem  66  toroidal motion i n 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 i n the diffusive regime, must be changed. T h e Lorentz force plays a more active role i n 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 B  z  is increased to ~ 20 G.  With  a sufficiently large Lorentz force, waves can propagate deeper i n the fluid before being attenuated. T h e solution will then be i n a wave regime, as we have seen i n the single plate solution. In figures 4.5 and 4.6, we have used the set of parameters presented i n table 4.1, but have replaced the magnetic field with 35 G. the magnetic field to 100 G. W i t h B  z  In figures 4.7 and 4.8, we have increased  = 35 G, the waves are sustained for many periods  of oscillation and can propagate deeper i n the fluid. T h e magnetic diffusion regions are extended and represent a good fraction of the flow i n the column. T h e amplitude of the waves is still smaller than the rigid rotation i n figure 4.5, but large enough to cause a significant deviation i n u from a rigid rotation. This confirms the results of the single plate analysis, which stated that the rigid rotations might not be an accurate description of the flow when the magnetic field is strong. A g a i n , this is consistent with the idea that the Proudman-Taylor constraint is broken by a strong magnetic field. This is even more evident i n figure 4.7 where B  z  = 100 G. T h e cur-  rent free region has been swallowed by the extended propagation of the waves. T h e 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  67  Conclusions  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. T h e 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 P r o u d m a n 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 i n the solid and a strong magnetic field. If the magnetic field is weak, the Elsasser number (3.8) is small and the solution stays i n a diffusive regime; rigid rotations will be generated even w i t h a strong conductivity i n the solid. If the conductivity of the solid is weak, a strong magnetic field will enable the wave to propagate but the amplitude of these oscillations w i 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 i n the fluid obtained w i t h the set of parameters listed i n table 4.1. T h e vertical axis of every graph represents the vertical distance z. T h e bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. T h e solution presented is at t = 0.  Chapter 4: Two-plate problem  69  B 4000  4000  1  ^  -1  0  1  « (G/m) x io"  7  o  1  (G/m) xio"  7  Figure 4.4: Solution of the scalar potentials of the magnetic field i n the fluid and the solid obtained w i t h the set of parameters listed i n table 4.1. T h e vertical axis of every graph represents the vertical distance z. T h e bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. T h e solution presented is at t = 0.  Chapter 4: Two-plate problem  u / u  70  A s  (m/s) xto"  Figure 4.5: Solution of the fluid velocity obtained with the set of parameters listed i n table 4.1, but substituting B by 35 G. The vertical axis of every graph represents the vertical distance z. T h e bottom fluid-solid interface is at z = 0, while the top fluid-solid interface is at z = 2000. T h e solution presented is at t = 0. z  Chapter 4: Two-plate problem  71  4000  4000  2000  0  5  (G/m) x 10"  7  2000  -5  0 (G/m) x i o '  5 7  Figure 4.6: Solution of the secondary magnetic field obtained w i t h the set of parameters listed i n table 4.1, but substituting B by 35 G. T h e 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. T h e solution presented is at t = 0. z  Chapter 4: Two-plate problem  U / U  72  A S  Figure 4.7: Solution of the fluid velocity obtained with the set of parameters listed i n table 4.1, but substituting B by 100 G. The vertical axis of every graph represents the vertical distance z. T h e bottom fluid-solid interface is at z — 0, while the top fluid-solid interface is at z = 2000. T h e solution presented is at t = 0. z  Chapter 4: Two-plate problem  -0.02  73  0.06  -200  0  (G/m)  2  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 i n 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 ( T O ) [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. T h e 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. T h e 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 , will generate waves that will 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. T h e purpose of this chapter is to investigate the transfer of angular m o m e n t u m 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 i n the fluid and neglect the coupling between  adjacent  cylindrical shells, the angular momentum balance relates the changes i n the angular velocity u(s) of the shell at radius s to the torques due to magnetic stresses —fi(s) and —f (s) at the I C B and C M B , respectively. W h e n the angular velocity is proportional to m  exp(—iu;t), the angular momentum balance is expressed as [Braginsky, 1970]  -u> m(s)u(s) = iwfi(s) + iuf (s). 2  m  (5.1)  T h e moment density, m(s), of a cylindrical shell of height h(s) is defined as  m(s) = ATvps h(s).  (5.2)  3  Since the magnetic torques are proportional to the relative motion between the and the solid, we can express fi(s) and f (s) m  fluid  as  / . ( , ) = T ^ ' ) [ « ( , ) - u<]  (5.3)  / « ( * ) = T ( * ) [*(*) - « J .  (5-4)  m  where u and u {  m  are the angular velocities of the inner core and the mantle. Buffett [1997]  developed the amplitudes  T (*) = « ( l + 8  i  and T , which can be expressed i n our geometry as: m  0B,V  /  (5.5)  Chapter 5: Transfer of angular momentum  76 (5.6)  T h e difference between these amplitudes arises because of the different conductivity m o d e l used at each boundary. It has been assumed that the conductivity of the mantle cr is m  confined to a layer of thickness A at the C M B . T h e thickness of this layer is assumed to be m u c h smaller than the typical magnetic skin depth i n the mantle. T h e inner core is assumed uniformly conductive with a value identical to that of the fluid cr,. T h e skin depth  T h e amplitude of the rigid velocity u reflects the angular m o m e n t u m transferred to the fluid by the solid. Isolating u(s) from equation (5.1) gives  B\ [(l + i)<r 6 + 4<r A] f  f  m  (5.7)  iAph(s}  which is readily evaluated i n terms of the parameters of our system; u and u i  m  are forcing  terms, which are equivalent to uf and uf i n the previous chapter.  5.3  Comparison of theories as a function of B  z  T h e 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 w i t h the angular momentum transferred using the model from the previous chapter, where rigid rotations are not assumed. Instead, a complete solution for the fluid m o t i o n is used to calculate the coupling. For weak magnetic field, we should expect little or no difference because our m o d e l 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  Chapter 5: Transfer of angular momentum  77  from that obtained using rigid rotations.. Comparisons are made for a range values of B . z  Thus, the departures in angular momentum transfer can be related to the departures from rigid rotations. In table 5.1 we present the other parameters used in the comparison.  p  1.1 x 10 kgm~ 47T x IO" kgmA~ s~ 3A x 10* Sm3.4 x 10 5 m 200 m 7.27 x I O * 2000 km ft/200 = 3.64 x I O " * . u>7r/180 = 6.34 x I O * 1 0 - W / 1 8 0 = 6.34 x I O s' 4  z  7  f  a  A ft  h  2  2  1  s  - 1  - 5  - 1  7  - 1  - 9  - 1  - 1 0  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 B . The solid line represents the z  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 B . We can imagine that the bottom .cylinder and z  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. T h e toroidal velocity of the column is only a fraction of that of the plate and out of phase by almost 180 degrees. A s 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 b o t t o m plate. For magnetic field not large enough to provide a perfect coupling (i.e. u/u  i  = 1) but  sufficiently large to ensure that the motion of the fluid cylinder is almost i n phase w i t h the motion of the bottom plate, the velocity of the fluid cylinder overshoots the velocity of the b o t t o m plate. T h e ratio u/u  i  is larger than 1 and this signifies that the inertia transmitted  to the fluid cylinder is large. T h e 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 rotations. T h e angular momentum transferred to the fluid is then identical to that obtained using rigid rotation. However, the magnetic force becomes important i n the force balance at roughly 20 G, and the departures with Braginsky's theory coincide w i t h the onset of waves which cause departures of the fluid from rigid rotations. For large magnetic  field,  though, the curve from m y model converges back towards the rigid rotation curve. A large magnetic field stiffens the fluid, prohibiting large amplitude motion. T h e 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.  79  Chapter 5: Transfer of angular momentum B o t h curves i n figure 5.1 have a similar shape, but the one from m y 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 B is required to obtain the same coupling. A n d as a result, there is a window of magnetic z  field values for which there is a significant difference i n the angular m o m e n t u m 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. T h e 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 w i t h 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 i n 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 i n the angular m o m e n t u m 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. T h e coupling is thus enhanced w i t h a longer period and, conversely, a shorter period would reduce the coupling. Similarly, the height of the fluid column will 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/u . i  80  Chapter 5: Transfer of angular momentum  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 B  z  = 50G. T h e other parameters are  the same as those presented i n table 5.1. Once again, the solid line represents the angular momentum transferred to the fluid w i t h the assumption of rigid rotations (equation 5.7), while the dashed line is the integrated velocity from the more complete model. T h e 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 b o t t o m plate is oscillated at high frequency (low period), the fluid column will not have time to adjust to the movement of the plate. T h e elastics are stretched over a time scale which is too short to overcome the inertia of the fluid cylinder. T h e amplitude of the toroidal motion of the fluid cylinder will only be a fraction of the amplitude of the plate and their motion will be out of phase.  W h e n 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 i n angular momentum. T h e dashed curve, representing the more complete model, is similar to the solid curve, obtained w i t h Braginsky's theory, but there is a. clear discrepancy between the curves. For a fixed J5 , my model requires a lower period to reach the same ratio uju Z  i  than that  required by the rigid rotation model. This reflects that the deformation i n 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 i n figure 5.4. W i t h B  z  = 50 G, m y model generates a solution where waves  are present (e.g. wave regime), so departures from rigid rotations are expected. However the departures i n angular momentum are confined to a range of frequencies.  For high  frequencies, the coupling is inefficient i n both models and the differences between them are not important. For very low frequencies, there is not enough inertia i n the fluid to generate waves w i t h amplitudes sufficiently large to break the rigid rotation. The plot of differences between the two models i n 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 will be m a x i m u m . Weaker magnetic fields will shift the position of the resonance towards longer periods and broaden the peak, while larger magnetic fields will shift the position of the resonance towards shorter periods and sharpen the peak. W h e n 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 i n 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 will occur is always defined by the value required to break the rigid rotations ( « 20 G).  T h e 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 will 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 w i d t h of the window is reduced. If the frequency is low enough that this  82  Chapter 5: Transfer of angular momentum  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; A t 10 000 days, the difference is negligible. A smaller B would displace the resonance towards longer periods, but eventually B becomes too z  z  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 momentum 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 5.5.1  Influence of waves on angular momentum transfer 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 i n the fluid column are much like the standing waves on an oscillating string. T h e frequencies and structure of these waves are constrained so that a finite number of half-wavelengths fit perfectly between the boundaries. T h e fundamental mode consists of a single half-wavelength and the higher modes are the harmonics of this fundamental mode. W h e n 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 i n figure 5.4 of the previous section. T h a t latter resonance is defined as the frequency at which the discrepancy between the two models is m a x i m u m . In this present section we are considering the excitation of standing waves on the fluid column. T h e effects of the standing waves are shown i n figure 5.5, which is the same graph presented i n figure 5.1, but extending the range of magnetic field values. For B  larger  z  than 120 G, the more complete model (dashed line) indicates rapid fluctuations i n 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 i n the mantle to A = 2000/sm. T h e period of oscillation is chosen to be 200 days as before, and we use u  m  (tt  m  = u  {  = c*>7r/180) to keep the solution as simple as possible. The other parameters used i n  the calculation are the same as those presented i n table 5.1.  84  Chapter 5: Transfer of angular momentum  Figures 5.6 and 5.7 show the results obtained using the two models as B is increased. z  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 (1 +  i)B\  3  (5.8)  HV2OJ  where 7/ is the magnetic diffusivity of the inner core or the mantle and r}f is the magnetic n  diffusivity in the fluid. We note that (5.8) reduces to (5.5) when n = Vf- Using (5.8) in n  the angular momentum balance (5.1) gives u, the angular velocity of the rigid rotations K  +  K  — iphoj  u (s)  = K  +  (5.9)  where B (l + i) 2  K =  z  The dashed line in figures 5.6 and 5.7 is again the result of the more complete modeL For B larger than 100 G, there are fluctuations due to the excitation of standing waves on z  the fluid column. A t 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 fundamental 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, will decrease the wavelength of the wave. We will thus hit the successive higher harmonics. A s the wavelength decreases, diffusion becomes more important, broadening the peaks and lowering the amplitude of the resonances.  This effect is most obvious on figure 5.6.  W h e n 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 i n figure 5.5 at a period of 200 days have disappeared at a period of 400 days i n figure 5.2. A s we have seen i n section 5.4, the amplitude of the waves i n 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 will also be limited. For decade periods, where there is not enough inertia i n the fluid to generate waves of significant amplitude, the frequency is too low to excite the resonances.  Hence, the  natural modes i n 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. T h e largest departures brought by the standing waves occur at very large magnetic fields, which are greater than the plausible values of the magnetic field i n the core.  Chapter 5: Transfer of angular momentum  5.5.2  86  Complications when the plate velocities are different  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 i n the toroidal velocity i n the fluid from one plate to the other, which cannot be described with a rigid rotation. T h e amplitude of the wave part of the solution increases i n order to satisfy the boundary conditions, and the amplitude of the standing waves increases. This effect is shown i n 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 i n order to satisfy the boundary conditions. The dashed line, which represents the more complete model, exhibits the resonant excitation of the standing waves. A g a i n , the discontinuity at the stronger magnetic field ( ~ 700 G) is the location of the fundamental mode, and the higher harmonics are found with decreasing  5.5.3  B. z  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, T h e average velocity i n m y model (dashed line) at large magnetic field does not converge to the velocity of rigid rotations obtained  87  Chapter 5: Transfer of angular momentum w i t h (5.9) (solid line). In the rigid rotation model, u at large field is closer to u than i  u  m  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 B  z  is varied. A larger magnetic  field will 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 0.55, which is the average value of the motion of each plate (uf  converges towards  = 1.0, uf  plate is contributing almost equally to the total torque on the fluid.  = 0.1). E a c h  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 will connect the velocity at one boundary .to the velocity at the other. T h e 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. B u t 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 will always be significantly stronger even for very large magnetic fields. O n the other hand, magnetic fields exceeding 100 G are at the limit 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 w i t h 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  1  _ I 04  20  0  1  1  40  60  1  J  80  :  1  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  0  ' 10  ' 20  ' 30  ' 40  ' 50  Bz  ' 60  1  70  ' 80  1  90 (G  1  100  >  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.  89  Chapter 5: Transfer of angular momentum  1.4  (Days)  Figure 5.3: Transfer of angular momentum to the fluid as a function of the period of oscillation for B = 50 G. T h e solid curve represents the toroidal velocity of rigid rotations obtained w i t h (5.7). T h e dashed curve is the average velocity from the results of m y model. T h e other parameters used i n the calculation are given i n table 5.1. z  0.9 ,0.8 0.7 -  CO  0.5 -  \ 1  0.6 -  3  0.40.3 0.2 0.1 -  1.1 I 0  '  100  '  200  '  300  '  400  T  '  500  '  600  '  700  '  800  •  900  1  1000  (Days)  Figure 5.4: Departures of our model from rigid rotations. This graph represents the difference between the two curves presented i n figure 5.3. T h e difference is reminiscent of a resonance phenomenon. We used B = 50 G, and the other parameters used i n the calculation are given i n table 5.1. z  Chapter 5: Transfer of angular momentum  90  Figure 5.5: Solution for the toroidal fluid velocity at large values of magnetic field. V a r i ations i n the amplitude of u with B reflect the resonant excitation of standing waves on the fluid column. The parameters used i n the calculation are given i n table 5.1 z  Chapter 5: Transfer of angular momentum  50  100  150  200  250  91  300  350  400  450  500  (G) Figure 5.6: Resonant excitation of standing waves on the fluid column (part I). T h e solid curve represents the toroidal velocity of rigid rotations obtained w i t h (5.9). T h e dashed curve is the average velocity from the results of my model. A p a r t from u = u- and A = 2000 km, the parameters used i n the calculation are given i n table 5.1. Refer to figure 5.7 for magnetic field values above 500 G. m  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 platevelocities (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 i n 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.  T h e previous chapter confirmed that rigid rotations i n 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. W h e n a cylindrical shell is rotated out of alignment w i t h the others, this will deform the radial magnetic field that threads the surface of the cylinders. B y Lenz' law, the deformation of this magnetic field will generate a magnetic force that opposes the deformation. T h e radial magnetic field provides a restoring force which permits oscillations between adjacent cylindrical shells. The dynamo of the E a r t h 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 w i t h 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 i n the column. W h e n fluid cylinders are differentially rotated, a magnetic torque is exerted on the fluid i n the disc by the distortion of the radial magnetic field. It is intuitive to think that only the region i n the vicinity of the disc will 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 w i t h height and hence, that rigid rotations can hardly be generated from a height dependent  B. s  However, this simple intuition does not take into account the effects of fluid circulation. T h e physical insights gained i n the previous chapters suggest that a torque confined to a narrow disc within the fluid column causes an adjustment i n the fluid i n the vicinity of the disc. We might then expect the presence of boundary layers. T h e radial transport i n those boundary layers would generate a radial transport i n the current free region to conserve fluid mass. B y angular momentum conservation, a rigid body rotation could be induced i n the whole column of fluid. T h e goal of this chapter is to demonstrate that we can indeed transfer angular momentum to the whole column of fluid when the torque is applied to a t h i n 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 B to estimate the torque, as Braginsky [1970] did. 3  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).  top  plale  upper  fluid  cylinder  fluid d i s c  >  l o w e r fluid c y l i n d e r  I  bottom  fluid  column  plalc  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 i n terms of the differential rotation of fluid cylinders. We assume that this torque can be captured by an azimuthal body force, F = F^e~ ' <j). This %u  t  body force is distributed across the whole thickness of the disc, so that the total force acting on the disc of thickness A is  = F^L\ . This force depends on the radial magnetic  2  Z  field B but the details are irrelevant for our purpose. 3  T h e body force  is included i n the momentum balance for the fluid within the disc.  T h e momentum and the induction equation i n the disc obey:  d dt  I  F  —v + v • Vv + 20 x v = -VTT + — ( V x B ) x B + —  (6.1)  J^B =  (6.2)  pu.  V  x (v x B) -  V  p  x (T/V x B)  We use the same field representation i n (2.5)  and eliminate the non-linear terms. T h e  von K a r m a n similarity solution eliminates the radial variable if the amplitude of the forcing is linearly dependent with s (e.g.  = sf^). Such a case would be expected i f the amplitude  of the radial magnetic field is increased w i t h radial distance s. T h e equations that govern the evolution of the scalar variables i n the disc are identical to those for the fluid cylinders except for the influence of  j/  - B.y,  |i dt !!^ dt dz  -V^P=0  - p[Lh.U dz  dz  + 2fi  2  i n the toroidal momentum equation:  (6.3)  =  A pA  JU-^!lp.O dz  (6.5) z  PH dz  3  (6.6)  97  Chapter 6: Rigid rotations from a concentrated torque 6.2.2  Influence of a prescribed force on a fluid column  There are several approaches for calculating the influence of a prescribed force  on the  One strategy would be to solve equations (6.3) to (6.6) subject to  motion of the fluid.  suitable boundary conditions on the upper and lower surfaces of the disc. However, we are not interested i n the evolution of the fields within the disc, especially when the disc is very t h i n . We are merely concerned w i t h the effect of a concentrated force on the remainder of the fluid column. T h e quantities that are essential for this purpose are the values of each scalar variable at the boundaries of the disc since these values provide boundary conditions for the solutions above and below the disc. A n alternative approach, if the disc is thin, is to integrate a solution across the disc to obtain j u m p conditions on the scalar variables on either side of the disc. In this way the solutions i n the two fluid cylinders can be related without requiring an explicit solution for the scalars inside the disc. We show below that the j u m p conditions depend on the amplitude of the applied force f^. This means that the effect of  is introduced through  non-homogeneous boundary conditions on the fluid solutions above and below the disc. Integration of (6.3) to (6.6) may be expressed i n terms of the average value of the scalars across the thickness A  of the disc. If we let {} denote the average value, the  z  integration yields  -iu{P}A  -  z  B  z  -uu{f }A  - B  -iu{u}A  - 2tt (i>  Z  +  z  - » {  (u  +  z  - £ * - )  +  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. T h e 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 t h i n these terms depend on A , which vanishes. We assumed that  is constant when A  z  the body force  increases as A  z  2  —> 0, so  —* 0 like a delta function.  Before evaluating the j u m p conditions, it is useful to interpret the terms i n equation (6.9), which may be viewed as an angular momentum budget for the fluid i n the disc. T h e first term on the left hand side represents the change i n the angular m o m e n t u m of the disc. T h e infinitesimal thickness implies that the disc carries a negligible amount of angular momentum. T h e second term involves the Coriolis force and relates to the transfer of fluid mass between the disc and the surrounding  fluid.  The j u m p (tp  +  — tp~) i n the axial velocity  on either side of the disc represents the net axial flow pumped into or out of the disc. This axial pumping will be accompanied by a radial transport, which redistributes angular momentum to the whole column of fluid. T h e t h i r d term involves the Lorentz force and expresses the angular momentum transferred to the fluid by the exchange of charges. The j u m p i n 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 i n the fluid column. T h e cross product between the background axial magnetic field and this radial current will generate a Lorentz force that will 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 t h i n 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  B o u n d a r y conditions  The effect of a concentrated force on the fluid cylinders is modelled by introducing j u m p 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 i n 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 i n 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. T h e toroidal velocity perturbation produced by this force will also be symmetric about the center plane of the disc as shown i n figure 6.2 B . T h e radial transport generated in the disc also has to be symmetric (figure cause an asymmetry i n this flow.  6.2 C ) simply because there is nothing to  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)  |"^ dz  (6.12)  +  " | ^ ~ = °dz  Similar symmetry conditions can be obtained for the induced magnetic field. The radial transport caused by  concentrates or spreads the B field fines apart. This perturbs the z  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:  -B (V-r) z  - V ( | p t - | p - ) J  =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 B field lines. A net.axial induced magnetic field z  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 j u m p (^fj- — ~f~~) is proportional to the j u m p i n the axial velocity. A d d i t i o n a l conditions arising from equations (6.8) and (6.10) have not been used because 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 i n the previous chapters.  The solution i n both fluid cylinders is given by equation (4.3).  T h e induced magnetic field i n each solid plate is described by equation (4.4) if we assume that the conductivity is uniformly distributed through the solid. T h e 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 = z ) are listed i n (6.11) to (6.16). We now have a d  system of 16 equations and 16 unknowns, with 3 forcing terms: the amplitude of motion of each plate and the forcing  6.3.1  at z =  z. d  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 (B  z  = 5 G)  Chapter 6: Rigid rotations from a concentrated torque is used i n the first example. The period of oscillation imposed by  103 is 10 000 days (pa  30 years). T h e disc is located i n the middle of the fluid column, separating it i n 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  T h e list  of parameters used i n the calculation is given i n table 6.1.  p p 0~  1.1 x 10 kgmX 10 kg m A' s" 4  47T  z  2  - 7  2  3.4 x l O ^ m 1 5 1 x nr Sm' 1 x IO" Sm' 7.27 x I O " * 2000 km  1  1  1  1 5  ft  5  h B on  1  5G  Z  ft/10  4  = 7.27 x  lO^s'  1  os  WTT/180 = 1.26 x  lO-  «f.  a;7r/180 = 1.26 x  IO-  u  OS  f<j> d  z  h • 10~  1 0  ^-  1  1 0  *"  1  Nm~ 1000 km 12  3  Table 6.1: Parameters used i n calculating the torque on the fluid column due to a concentrated force.  Figures 6.3 and 6.4 show the profiles of the scalars u, rp, P, and T for this calculation. T h e 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 i n the previous chapters. T h e adjustment i n 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 i n the fluid decreases away from the position where the force is applied. T h e oscillating behavior i n this boundary layer is governed by the same processes that apply i n the fluid-solid boundary layers. T h e main difference resides i n the couphng mechanism. In the case of the fluid-solid couphng, the adjustment i n the fluid was necessary to  Chapter 6: Rigid rotations from a concentrated torque respond to the motion of the plate, which carried the B  z  104  field lines w i t h 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 i n the disc by the concentrated force and hence, no charges exchanged between the fluid and the disc. T h e force is communicated to the fluid cylinders through a transfer of fluid mass, which amplitude is proportional to f^. T h e adjustment i n the fluid is not initiated by a toroidal acceleration i n the boundary layer, as it was the case for the coupling at the fluid-solid boundaries. T h e transfer of fluid mass out of (or into) the disc implies a net radial transport i n 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 i n the fluid. T h e poloidal magnetic field i n the boundary layer is associated w i t h a toroidal electric current (see figure 3.1 C ) . T h e Lorentz force results from the interaction of this current w i t h B  z  and accelerates the fluid  i n the radial direction. T h e adjustment of the fluid i n the boundary layer region is thus initiated by a radial acceleration. Moving  to a different height will produce the same rigid rotation, as the angular  momentum i n the fluid column is proportional to  and is independent of its position. T h e  amplitude of this rigid rotation is out of phase by 90 degrees w i t h ±(f^/pui)  and varies between  ( ± 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 w i t h the same set of parameters used i n figures 6.3 and 6.4 but w i t h a magnetic field of B  z  = 50 G. A g a i n , the  results are similar to those observed earlier with electromagnetic coupling at the fluid-solid boundary. T h e 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 i n 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 t h i n boundary layer. Hence, we can expect that the solution will 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 m a i n goal of this chapter is to demonstrate that the angular momentum transferred to the fluid column i n response to a concentrated torque is redistributed over the whole length of the fluid column. T h e 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 i n the L O D . We have selected a magnetic field of 30 G i n order to reduce the damping of the waves and increase the thickness of the boundary layers. This will facilitate the comparison between the boundary layers at the fluid-solid interfaces w i t h those at the position of f^. T h e list of parameters used i n the calculation is presented i n table 6.2. In figures 6.7 and 6.8, we present the solution i n both fluid cylinders obtained w i t h this  Chapter 6: Rigid rotations from a concentrated torque  106  1.1 x 10 kgm47r x I O kgm A' s~ 3.4 x 10 Sm' 3,4 x 10 5 m 3.4 x - l O ' S m " 7.27 x l O - ^ 2000 km 30 G ft/10 = 7.27 x IO" *WTT/180 = 1.26 x . l f r * IO" WTT/180 = 1.26 x IO" s' h-lQ- Nm~ 1000 km 4  p  3  -7  0~  °i  2  2  5  1  5  1  1  ft  5  h  4  U) .  1  9  1 0  ui  1  ul  Zd  - 1  11  12  f(j>  1  1  3  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 B to estimate the electromagnetic couphng between adjacent s  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  070726  0.7073  0.70734  -5  107  0  5  (m/s) x10"  7  Figure 6.3: Solution for the scalar potentials of the velocity for the set of parameters i n table 6.1. T h e bottom pate is at z = 0, the top plate is at z = 2000, and is at z = 1000. T h e superscripts A and B refer to the solution i n 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  1  (G) X10'  2  (G/m) x 10"  12  Figure 6.4: Solution for the scalar potentials of the magnetic field for the set of parameters i n table 6.1. T h e bottom pate is at z = 0, the top plate is at z = 2000, and is at z = 1000. T h e superscripts A and B refer to the solution i n 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. A p a r t from B = 50 G , the parameters used i n the calculation are given i n table 6.1. T h e bottom pate is at z = 0, the top plate is at z = 2000, and is at z = 1000. T h e superscripts A and B refer to the solution i n the fluid cylinder below and above the concentrated force. z  Chapter 6: Rigid rotations from a concentrated torque  Figure 6.6: Solution for the scalar potentials of the magnetic field. Apart from B = 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. z  Chapter 6: Rigid rotations from a concentrated torque  0.5  ~ A-  ~ A  111  0  0.5 A  (m/s)  1 xio"  6  u / u<  0.45  0.5  . 0.55  0.6  0.65  0.7'  (m/s) x 1 0  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  -  2  -  1  0  -4  1  (G) x 10"  5  .  -2  112  0  2  4  (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 T h e purpose of this work is to develop an improved model for electromagnetic coupling between the fluid core and the rest of the E a r t h . A good description of the exchange of angular momentum is required i n order to explain the observed variations i n the length of the day ( L O D ) . This work is motivated by potential problems that arise w i t h electromagnetic coupling inside the tangent cylinder. Existing models rely on the assumption that fluid motions i n 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 i n core angular momentum and the observed changes i n the length of day is encouraging, but it is important to assess the validity of the underlying approximations, especially inside the tangent cylinder. T h e calculation i n this thesis confirm that the assumption of rigid rotations is valid at decade periods or when the magnetic field is weak (e.g. B  z  <C 20 G).  However, when  the magnetic field is strong, the Proudman-Taylor theorem does not apply and the rigid rotations are disrupted. T h e magnetic stress computed at the boundary w i t h the velocities based on rigid rotations yields erroneous results since these velocities do not accurately represent motion at the fluid-solid interface, on which the magnetic stress depends.  It  is plausible that the magnetic field at the inner core boundary ( I C B ) is large enough to break the rigid rotations. Consequently, electromagnetic coupling for the cylindrical shells located inside the tangent cylinder has to be reassessed.  Chapter 7: Conclusions  114  T h e model developed i n this work is used to examine the behavior of the flow i n response to an electromagnetic coupling at the solid boundaries. T h e magnetic force, or Lorentz force, is retained i n the force balance. Rigid rotations are not assumed a priori i n the calculation of electromagnetic couphng. Instead the fluid motions are obtained by solving the momentum equation along w i t h the induction equation for the evolution of the magnetic field. T h e solution depends on a number of parameters, including the conductivity of the solids and the strength of the magnetic field. T h e relative importance of the forces and the strength of the electromagnetic couphng at the boundaries can be readily adjusted i n the calculation and this makes the model very versatile. In particular, the period of oscillation of the solid boundaries can be varied to assess the angular m o m e n t u m exchange between the core and the mantle at both short periods and long periods. This is important because variations i n the length of day are observed at both annual and decadal periods. T h e fluid core is modelled as an infinite layer of fluid bounded by two infinite plates. T h e fluid and the plates are all conducting and an ambient magnetic field, which is perpendicular to the fluid-solid interfaces, permeates the whole geometry. Inside the tangent cylinder, one plate represents the inner core and the other one represents the mantle. Outside the tangent cylinder, both plates have the characteristics of the mantle. T h e system is rotating at the frequency of rotation of the E a r t h and the axis of rotation is aligned w i t h the magnetic field.  The rotation of the system motivates the use of a cylindrical  coordinate system. T h e velocity and magnetic fields are developed i n terms of toroidal and poloidal scalar potentials, and the infinite extent of the fluid i n the radial direction permits similarity solutions. Simple calculations using a single plate below an infinitely deep fluid identify the basic physical processes involved i n electromagnetic couphng between the fluid and sohd. According to the frozen-flux approximation, the plate carries the background magnetic field  Chapter 7: Conclusions  115  lines w i t h its motion and a shear i n the fluid velocity deforms the initial magnetic field into the direction of the fluid motion. This secondary magnetic field due to deformation is toroidal, and it is associated with an induced electric current, which is poloidal.  The  interaction of this poloidal current with the background magnetic field produces a toroidal force i n the fluid i n the direction of the plate movement.  Hence, the fluid tends to be  dragged by the motion of the plate. A n oscillation i n the rotation of the plate generates a wave i n the fluid. W h e n the magnetic field is small, the diffusion of the magnetic field perturbation is important and this wave is rapidly attenuated. A distinct boundary layer region develops i n this case as the fluid adjusts to the moving boundary. T h e magnetic field at the  fluid-solid  interface also diffuses i n the solid and is also  associated w i t h electric currents near the boundary. A key feature of the electromagnetic couphng is the communication between the current i n the solid and the current i n the boundary layer of the  fluid.  A n axial current across the boundary causes an exchange  of charges between the fluid and the sohd.  This axial current is deflected into a net  radial current i n the fluid i n order to conserve charge. We have previously noted that this poloidal current interacts with the background axial magnetic field to produce the toroidal Lorentz force that accelerates the fluid i n the boundary layer. Consequently, the angular momentum transfer from the sohd to the fluid may be viewed as a consequence of the exchange of charge by the axial current. Another key feature of the couphng is the radial fluid transport generated i n the boundary layer. This radial transport arises from the Lorentz force i n the fluid. A radial transport redistributes angular momentum since fluid parcels imported from a different radius change angular velocity to conserve their angular momentum. Hence, accelerations of the fluid i n the boundary layer are accomplished by. the joint effort of the Lorentz forces and radial transport. B y conservation of fluid mass, radial transport i n the boundary layer  Chapter 7: Conclusions  116  produces an axial flow out of the boundary layer, which pumps fluid from the boundary layer into the current free region (the bulk of the  fluid).  T h e introduction of the second plate restores the original geometry that better approximates the angular momentum transfer i n the core. T h e fluid cylinders are bounded by two conducting solids. T h e electromagnetic coupling occurs at both boundaries. W h e n there is an exchange of charge between the solid and the fluid, angular m o m e n t u m is transferred from the solid to the boundary layer regions i n the fluid by electromagnetic forces since this is where the radial currents are confined. W h e n the current at each boundary results i n a net transfer of charge between the two solids and the fluid (i.e. when the radial current at each boundary layer is not equal and opposite), then a net torque is imposed on the fluid column. T h e Lorentz force i n the boundary layer at both ends produces a net radial transport and the associated axial pumping. Unless the net radial currents i n the two boundary layers are equal and opposite, the axial pumping out of the boundary layer at one end will not be equal to axial pumping into the boundary layer at the other end. Hence, a net amount of fluid mass is transferred from the boundary layers into the current free region. To conserve fluid mass, a radial transport develops through the whole current free region.  B y conservation of angular  momentum, fluid parcels i n the current free region accelerate as rigid cylinders. T h e net angular momentum transferred to the fluid column is then not restricted to the boundary layers, but is distributed over the whole length of the cylinder by fluid circulation. T h e fluid circulation can also transfer angular momentum to the entire column when a localized torque is applied at a prescribed height. If the torque is confined to a narrow disc i n the fluid column, a net radial transport through the disc is driven by the applied torque to conserve angular momentum. This implies a mass exchange between the disc and the  Chapter 7: Conclusions  117  rest of the fluid column. T h e resulting circulation allows the whole column to accelerate even though the apphed torque is confined to a particular height on the fluid column. T w o distinct regimes of solution i n the fluid are identified. One is a diffusion regime while the other is a wave regime. For a fixed rotation rate, density and conductivity, the sole parameter that determines the regime of the solution is the strength of the background axial magnetic field. W h e n the magnetic field is weak, the solution is i n a diffusion regime. T h e restoring Lorentz force does not play an important role i n the force balance and the perturbation of the magnetic field induced at each fluid-solid boundary is rapidly attenuated. A w a y from the boundary, the electric currents vanish and the fluid is accelerated as a rigid body. T h e amphtude of the toroidal velocity i n the boundary layers is comparable to the amplitude of the rigid toroidal velocity of the current free region. Thus, the toroidal motion of the whole fluid column of fluid can be approximated as a rigid rotation. This result is i n agreement w i t h the Proudman-Taylor theorem, which states that the geostrophic fluid motion do not vary i n the direction of rotation (e.g. (ft • V ) v = 0). T h e Lorentz force is insufficient to break the Proudman-Taylor constraint when the magnetic field is weak. Since the perturbation i n the toroidal velocity i n the boundary layer is small, the toroidal velocity i n the fluid at the boundary can be assumed to be the velocity of the rigid rotations. T h e magnetic stress at the boundary may be computed from the velocity difference between the sohd and the rigid rotation velocity. This was the basis of Braginsky's model [1970]. We have obtained the angular momentum transferred between the solid and the fluid using, this assumption, and compared it w i t h the transfer obtained w i t h our electromagnetic couphng model that includes the perturbed velocities and currents. We confirm that our electromagnetic couphng model gives the same solution as Braginsky's  Chapter 7: Conclusions  118  model for the electromagnetic coupling i n the diffusive regime. W h e n the magnetic field is large enough to make the Elsasser number approximately one (at B  z  ~ 20 G), the interaction between this magnetic field and the current induced  by the perturbation at the boundaries becomes an important restoring force. This force enables the waves to propagate into the fluid away from the disturbance induced by the oscillation of the plates. This behavior is characteristic of the wave regime. T h e perturbed fields extend from the boundaries to the point where they overlap; the toroidal perturbation of the magnetic field occurs everywhere i n the whole fluid column and the current free region disappears. T h e fluid motions are no longer characterized by rigid rotations. This is consistent w i t h the idea that large magnetic fields generate large Lorentz forces which break the Proudman-Taylor constraint. T h e amplitude of the disturbance i n the wave regime depends on frequency. W h e n the frequency of oscillation of the plate is high, the efficiency of the coupling is low. T h e angular momentum transferred to the fluid is limited and the amplitude of the fluid disturbance is small. A t very low frequencies, however, there is simply not enough inertia to generate waves of significant amplitude. There is a window of frequencies between these extremes where the coupling is efficient and the inertia is large enough to generate waves of significant amplitude. For frequencies outside this window, the response of the fluid to the electromagnetic coupling with the oscillating boundaries can be approximated by rigid rotations even when the magnetic field is large enough to permit waves i n the solution. T h e window of frequencies where the fluid departs from rigid rotations varies w i t h the strength of the magnetic field. A weaker magnetic field shifts the resonance towards longer periods of oscillations. However, i f the magnetic field is weaker than « 20 G , the Elsasser number becomes less than one and the consequences of the Proudman-Taylor constraint  Chapter 7: Conclusions  119  act to suppress the departures from rigid rotations. Conversely, a stronger magnetic field shifts the resonance to higher frequencies.  B u t then again, a very large magnetic field  prohibits large deformations of the fluid and enforces rigid rotations. W h e n rigid rotations are not valid, the toroidal velocity at the boundary can be very different from the average toroidal velocity i n the fluid. Hence, the electromagnetic couphng can not be computed using the rigid rotations velocity to define the j u m p i n velocity at the boundary. We have confirmed this by showing that the departures from rigid rotations in our model correspond to departures i n the transfer of angular m o m e n t u m when the assumption of rigid rotations is used. These results were then applied to the context of the Earth's core. For the cylinders outside the tangent cylinder, the solution is probably i n a diffusion regime since the magnetic field at the C M B is weak. R i g i d rotations remains valid even at annual periods. For the cylinders inside the tangent cyhnder, the magnetic field is believed to be large at the I C B and the toroidal oscillations of the inner core will generate a wave that will disrupt rigid rotations. This modifies the idea that strong couphng at the I C B will force the whole tangent cyhnder, including the inner core, to rotate as a rigid body. For the decade period of oscillation i n the length of day ( L O D ) , however, there is not enough inertia to create large departures from rigid rotations. T h e couphng at the I C B will be efficient and the whole tangent cyhnder should rotate as a rigid bulk. In this work, we have also confirmed that electromagnetic couphng between adjacent cylindrical shells can be calculated by assuming that the torque is uniformly distributed over the surface of the shells.  This was done by showing that the angular m o m e n t u m  transferred to the fluid column i n response to a torque apphed at a particular height was redistributed to the whole length of the fluid column. Our results show that rigid rotations  Chapter 7: Conclusions  120  arise from localized torques under the same conditions as for the electromagnetic couphng w i t h the boundaries. Hence, at decade periods, Braginsky's electromagnetic couphng model that rehes on rigid rotations is sufficient to correctly assess the transfer of angular m o m e n t u m between the mantle, the fluid core and the inner core. A magnetic field of 20 — 30 G at the I C B is suggested by recent dynamo calculations [Glatzmaier and Roberts 1995, K u a n g and B l o x h a m 1997].  A t this field strength, the  resonance is near the annual period of oscillation i n the L O D . T h e electromagnetic couphng computed assuming rigid rotations will be inaccurate inside the tangent cyhnder. Hence, i f one wants to assess the contribution of the core i n the angular momentum balance at short periods, rigid rotations i n the fluid core can not be used to determine the electromagnetic couphng. More accurate models for the toroidal velocity perturbation based on the work of this study must be used.  121  References Braginsky, S. L , 1970, Torsional magnetohydrodynamic vibrations i n the Earth's core and variations i n day length, Geomag. Aeron., 10, 3-12 (Engl. Transl. 1-8) Buffett, B . A . , 1992, Constraints on magnetic energy and mantle conductivity from the forced nutations of the E a r t h , J. Geophys. Res., 97, 19581-19597 Buffett, B . A . , 1996, Gravitational oscillation i n the length of day, Geophys. Res. Lett., 23, 2279-2282. Buffett, B . A . , 1997, Free oscillation i n the length of day: Inferences on physical properties near the core-mantle boundary, i n The core-mantle boundary region, edited by M . Gurnis, Geodynamics monograph, American Geophysics U n i o n , Washington, D . C ,  In press. Chao, B . F . , 1994, Transfer function for the length-of-day variations: Inference for E a r t h dynamics (abstract), Eos Trans. A G U , 75, Spring meet, suppl., S i l l . Glatzmaier, G . and P. H . Roberts, 1995, A three-dimensional convective dynamo solution w i t h rotating and finitely conducting inner core and mantle, Phys. Earth Planet. Inter., 91, 63-75. Glatzmaier, G . and P. H . Roberts, 1996, Rotation and magnetism of the Earth's inner core, Science, 274, 1887-1890. Greenspan, H . P. and L . N . Howard, 1963, O n a time-dependent motion of a rotating fluid, / . Fluid Mech., 17, 385-404. Gubbins, D . and P. H . Roberts, 1987, Magnetohydrodynamics of the Earth's core, i n Geomagnetism, vol 2, ed. J . A . Jacobs, Academic Press, New York. Hide, R . , 1969, Interactions between the Earth's liquid core and mantle, Nature, 222, 1055-1056. Jackson, A . , J . B l o x h a m and D . Gubbins, 1993, Time-dependent flow at the core surface and conservation of angular momentum i n the coupled core-mantle system, i n Dynamics of Earth's Deep Interior and Earth's Rotation, edited by J . - L . Le M o u e l , D . E . Smylie and T . Herring, pp. 97-107, Geophysical Monograph 72, I U G G Volume 12, 1993.  References  122  Jault, D., C. Gire and J.-L. Le Mouel, 1988, Westward drift, core motions and exchange of angular momentum between core and mantle, Nature, 333, 353-356. Jault, D., C. and J.-L. Le Mouel, 1989, The topographic torque associated with tangentially geostrophic motion at the core surface and inferences on the flow inside the core, Geophys. Astrophys. Fluid Dyn., 48, 273-296. Kuang, W . and J . Bloxham, 1993, The effect of boundary topography on motions of the Earth's core, Geophys. Astrophys. Fluid Dyn., 72, 161-195. Kuang, W . and J . Bloxham, 1997, A n Earth-like numerical dynamo model, Nature, 389, 371-374. Langel, R. and R. Estes, 1985, The near-Earth magnetic field at 1980 determined from Magsat data, J. Geophys. Res., 90, 2495-2509. Loper, D . E . , 1970, Steady hydromagnetic boundary layer near a rotating, electrically conducting plate, Phys. Fluids, 13, 2999-3002. Loper, D . E . , 1971, Hydromagnetic spin-up of a fluid confined by two flat electrically conducting boundaries, J. Fluid Mech., 50, 609-623. McCarthy, D . D . and A . K . Babcock, 1986, The length of day since 1656, Phys. Earth Planet. Inter., 44, 281-292. Ponte, R. M . , 1990, Barotropic motions and the exchange of angular momentum between the oceans and sohd Earth, J. Geophys. Res., 95, 11369-11374. Pedlovsky, J., 1986, Geophysical Fluid Dynamics, 2nd E d . Springer Verlag, New York. Proudman, J . , 1916, On the motions of solids in a liquid possessing vorticity, Proc. R. soc. Lond., A92, 408-424. Roberts, P. H . and S. Scott, 1965, On the analysis of the secular variation. I. A hydromagnetic constraint: theory, J. Geomagn. Geoelect., 17, 137-151. Rosen, R. D., 1993 , The axial momentum balance of the Earth and its fluid envelope, Surveys in Geophysics, 14, 1-29. Stephenson, F . R. and L . V . Morrison, 1984, Long-term changes in the rotation of the Earth: 700 B . C . to A . D . 1980, Phil. Trans. R. soc. London, A313, 47-70. Taylor, G . I., 1917, Motions of solids in fluids when the flow is not irrotational, Proc. R. soc. Lond., A93, 99-113. Taylor, J . B . , 1963, The magneto-hydrodynamics of a rotating fluid and the Earth's dynamo problem, Proc. R. soc. London, A274, 274-283.  References  123  Zatman, S. and J . Bloxham, 1997, The phase difference between the length of day and atmospheric angular momentum at subannual frequencies and the possible role of core-mantle couphng, Geophys. Res. Lett., 24, 1799-1802.  

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