The inner core as a dynamic viscometer

Abstract

The inner core is likely surrounded by a slushy layer consisting of both solid and liquid phases. We show that the direct measurement of the viscosity in this region is possible using the observed reduction in Coriolis splitting of the two equatorial translational modes of oscillation of the inner core about its central position. In this method, the inner core itself becomes a dynamic extension of the traditional falling ball viscometer used in the laboratory. We have developed the Ekman boundary layer theory for the translational modes and made use of novel solutions of the Poincaré equation for the flow exterior to the boundary layer. The free constants in the exterior flow solutions are determined by continuity of radial displacement at the two boundaries and conservation of linear momentum between the inner core, outer core and shell. We are able to obtain analytic expressions for both the pressure and the viscous drags for a sphere oscillating in a contained rotating fluid. The pressure and viscous drag expressions allow us to find new splitting laws for the three translational modes. The effect of viscosity is more complex in this case than in the usual dashpot damping of a harmonic oscillator. In addition to the increase in apparent inertia of one-half the displaced mass found in the classical literature on ideal fluid dynamics, viscosity contributes to terms in the square of the angular frequency (coupled inertia), linear in angular frequency (prograde–retrograde splitting), and terms independent of angular frequency (alteration of the apparent restoring force). The real part of the equation of motion for the inner core yields a general splitting law from which a splitting diagram is obtained. Although the axial mode splitting curve is not much affected, the splitting of the two equatorial modes is reduced by viscosity. A single viscosity of 1.22×10 11 Pa s, near the upper limit of 10 11 Pa s for the bulk viscosity derived theoretically by Stevenson (1983) for two-phase fluids, reduces the splitting of both equatorial modes to the observed periods. The viscosity is easily obtained from the observed equatorial periods with a precision of 1%.