Viscosity of Earths Outer Core

arXiv:0709.3333v1 [physics.geo-ph] 21 Sep 2007 Viscosity of Earth’s Outer Core† D. E. SMYLIE1 and Andrew Palmer2 1Department of Earth and Space Science and Engineering, York University 4700 Keele Street, Toronto, Ontario, M3J 1P3, CANADA Phone:(416) 736-2100, ext. 66438, Fax:(416) 736-5817 E-mail: doug@core.yorku.ca 2Department of Physics and Astronomy, York University 4700 Keele Street, Toronto, Ontario, M3J 1P3, CANADA E-mail: palmer@core.yorku.ca Abstract A viscosity profile across the entire fluid outer core is found by interpolating between measured boundary values, using a differential form of the Arrhenius law governing pressure and temperature dependence. The discovery that both the retrograde and prograde free core nutations are in free decay (Palmer and Smylie, 2005) allows direct measures of viscosity at the top of the outer core, while the reduction in the rotational splitting of the two equatorial translational modes of the inner core allows it to be measured at the bottom. We find 2, 371 ± 1, 530 Pa · s at the top and 1.247 ± 0.035 × 1011 Pa · s at the bottom. Following Brazhkin (1998) and Brazhkin and Lyapin (2000) who get 102 Pa · s at the top, 1011 Pa · s at the bottom, by an Arrhenius extrapolation of laboratory experiments, we use a differential form of the Arrhenius law to interpolate along the melting temperature curve to find a viscosity profile across the outer core. We find the variation to be closely log-linear between the measured boundary values. The close agreement of the boundary values of viscosity, found by Arrhenius extrapolation of laboratory experiments, with those found from the free core nutations, and the inner core translational modes, suggests that core flows are laminar and that the returned viscosities are measures of their molecular values. This would not be the case in the presence of the vigourous turbulent convection sometimes postulated by dynamo theorists. The local Ekman number is found to range from 10−2 at the bottom of the outer core to 10−10 at the top. Except in the very lower part of the outer core, Ekman numbers are in the range 10−4 to 10−5, or below, in which the laminar flows of numerical dynamos and laboratory rotating fluids experiments occur. We find explicit expressions for the reciprocal Q’s at both boundaries and for the viscous cou- pling torques between the outer core and shell, and between the outer and inner cores. For the high viscosity in the F-layer outside the ICB found from the reduction in the rotational splitting of the two equatorial translational modes of the inner core, the inner core is found to be tightly coupled to the outer core with a negligible contribution to dissipation in the free core nutation modes. 0†Published electronically in arXiv.org>physics >physics.geo-ph, Cornell University Library, Ithaca, N. Y., August 2, 2018. 1 1 Introduction Properties of Earth’s deep interior, such as its elasticity, density, pressure and gravity have tradition- ally been obtained through the inversion of seismic observations. While these have been important to our understanding of Earth’s internal structure, the viscosity of the outer fluid core is crucial to our understanding of its dynamics and the generation of the geomagnetic field. Direct observations and limits on viscosity have traditionally been much larger than those values found on the basis of extrapolations of laboratory high pressure and temperature experiments (Lumb and Aldridge, 1991). The latter tend to be close to that of liquid iron at atmospheric pressure (Rutter et al., 2002), while the former are many orders of magnitude larger (Davis and Whaler, 1997). Unusual properties of the lower outer core have long been suspected, going back to the 1926 claim by Jeffreys (Jeffreys, 1926) of a strong negative P-wave velocity gradient there. While the P gradient is now thought to be small, but slightly positive, possibly due to solid inclusions slowing compressional waves in that region (Garland, 1971, pp.42-50), its properties remain a subject of speculation. A new seismic phase, PKhKP, has even been suggested by Bolt (Bullen and Bolt, 1985, p.317) as originating from reflections in the lower outer core. With the discovery that both the retrograde and prograde free core nutations are in free decay (Palmer and Smylie, 2005), direct measures of the viscosity at the top of the outer core can be made. In this paper, we present a detailed analysis of both the Ekman layer at the top of the outer core and that at the bottom, just outside the inner core boundary. Our analysis yields a mean value of 2, 371 ± 1, 530 Pa · s for the recovered dynamic viscosity at the top of the outer core. From the reduction in the rotational splitting of the two equatorial translational modes of the inner core, two independent measures of viscosity in the F-layer at the bottom of the outer core are found (Smylie, 1999; Smylie and McMillan, 2000), as the reduction is larger for the retrograde mode than for the prograde mo

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