J.P. Vanyo
Mechanical Engineering and Geological Sciences, University of California, Santa Barbara, CA USA.
We compare flows at and near the CMB inferred from observed geomagnetic secular variation including a generally accepted westward drift, experiments that precess a spinning liquid-filled oblate spheroidal cavity causing misalignment of the cavity and liquid spin axes, and predictions using equations of motion. Analyses, beginning with Poincare, introduce a solution of liquid motion in a precessing cavity, equivalent to the first term in a series expansion, as a rigid body component of motion. Tentative acceptance of this first term as representing the entire solution for a nearly spherical cavity leads to solution of a rigid sphere interior to and coupled to the cavity wall. Relative motion and energy dissipated are computed and verified in experiments to be accurate within several percent. Relative motion and energy dissipation experiments have achieved Poincare numbers of 10(-6) compared to the earth’s 10(-7) and Ekman numbers on the order of 10(-7). Oblate ellipticity is 1/400 as for the CMB. The experiments also illustrate internal motion driven by some combination of inertia waves and surface Ekman pumping. Previous analyses of core-mantle relative motion by Vanyo and associates and by Pais and Le Mouel are continued as follows. A coordinate system with its equator coincident with the earth’s and a prime meridian fixed in the direction the core axis lags the mantle axis in precession is used. The core-mantle relative motion pattern is fixed in this space, and the mantle and core rotate daily, nearly in unison, through this pattern. A vector representing angular velocity of the core relative to the mantle is fixed in the direction of the prime meridian and, in general, will lie at an angle ‘beta’ south of the equator. Its rotation sense indicates secular westward drift of the core. The intersection angles between polar (latitude) circles of the angular velocity vector and meridians of the prime coordinate system provide relative translational velocities of the core and mantle at each latitude as a function of longitude as the core and mantle rotate. Core-mantle relative motion at the equator, over time, approximates a sine wave. A secular westward drift is superimposed on a north-south oscillation caused by angular displacement of core and mantle equators. At latitude beta, the poleward tips of the north-south pattern become cusps. At higher latitudes, the ‘tips’ fold over to become anti-cyclonic near-elliptical loops transforming to near-circles at polar latitudes. A core-mantle lag angle of 35 arc seconds causing an approximate displacement of the axes at the CMB of some 600 m and an average daily westward drift at the equator of some 40 m/day has been estimated by Vanyo and associates to fit known data reasonably well. The equatorial ‘sine-wave’ motion then has 1.2 km amplitude and 40 m wavelength. Motion at higher latitudes always has 1.2 km amplitude (and finally that diameter) but a lesser westward drift equal to 40 m times cosine of latitude. Comparison to observations is complicated in that at each latitude the pattern overlaps with different patterns at adjacent latitudes and with the same patterns to the east and west that are displaced in phase, and further, one sees only the net result after some thousands of cycles, e.g., 10 yr. times 365 days or 20 minutes times 200 rpm. An observer will see an effect similar to that obtained by translating a sheet of glass in an elliptical or circular pattern over a flat surface covered with sand or coarse grease. The glass translates in that pattern; it does not rotate; and the motion is small compared to the area of the glass. For northern hemisphere simulation, the center of clockwise motion moves slowly left. Observation is greatly augmented by introduction of a tracer fixed to the fluid, e.g., an injection of dye in an experiment or a magnetic field anomaly at the CMB. The Vanyo and Dunn experimental paper shows surface patterns of dye advection only at fixed times and after thousands of cycles. Real time observations of the rate and direction of advection are not yet available in the published literature. The photographs and real time observations are consistent with calculations of a rigid-sphere model. Secular motion of earth surface geomagnetic anomalies have been used to estimate core-mantle relative motion, e.g., by Voorhies, by Matsushima, and by Holme and Whaler. Their estimates use the property that a magnetic anomaly will remain fixed to the same particles of any fluid with high electrical conductivity. Vortices in a nearly invisid fluid have the same property of remaining with the same fluid particles. Parallel vortices with the same rotation sense have the additional property that they will rotate each other about their centroid in much the same way the earth and moon orbit about their common mass center. A small viscosity will cause the separate vortices to coalesce into one large vortex. A distribution of small diameter relative motion loops generated at the CMB over thousands of days will have random encounters as they move westward and begin to coalesce into larger and larger vortices. Magnetic field anomalies carried by these larger vortices will be seen as magnetic vortices moving slowly westward with anti-cyclonic rotation. Both Voorhies and Matsushima calculate flow patterns similar to the above, indeed, both show one large anti-cyclonic vortex in each hemisphere combined with lesser vortices, some cyclonic some anti-cyclonic. Both also show continuous westward flow at the equator. In that magnetic observations may not see deeper than a conducting layer in the lower mantle, their analyses may illustrate how field anomalies are dragged through the lower mantle by a field fixed in the core. Proposed core-mantle coupling mechanisms have included pressure forces due to ellipticity, viscosity through a laminar boundary layer or modified by a turbulent layer, topographic forces due to CMB surface irregularities, and coupling by a magnetic field traversing the CMB to a conducting lower mantle. We find no conclusive evidence, theoretical or experimental, to discount any of these mechanisms. All may contribute to coupling, but the imprint of core flow patterns in the surface magnetic field does require that magnetic coupling be a significant, and perhaps dominant, portion of the total. For a paper in progress, with diagrams and photographs, see: http://www.me.ucsb.edu/dept_site/vanyo.htm