Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 480

The boundary values of viscosity we have found are in very close agreement with
an Arrhenius extrapolation of laboratory experiments by Brazhkin (1998) and by
Brazhkin and Lyapin (2000), who find 10 11 Pa s at the bottom of the outer core and
10 2 Pa s at the top. We are prompted, by the very close agreement of the viscosity
measures at the boundaries with those provided by the Arrhenius extrapolation, to
interpolate between the boundary values in order to obtain a viscosity profile across
the entire liquid outer core.
The Arrhenius description of the temperature and pressure dependence of the
dynamic viscosity η is (Brazhkin, 1998)

exp E act 0 +

PV act
kT

η ∼ , (8.126)

with E act 0 representing the activation energy at normal pressure, P the pressure, V act
the activation volume, k Boltzmann's constant and T the Kelvin temperature. V act
is proportional to the atomic volume, which in turn is inversely proportional to the
density ρ. While the activation volume for liquid metals at atmospheric pressure is
very small, Brazhkin (1998) and Brazhkin and Lyapin (2000) report experimental
results on pure iron at the melting temperature, T m , that show it to be strongly
dependent on pressure up to 95 kbar. The strong pressure dependence requires
integration of the di erential form of the Arrhenius expression. For dominant pres-

sure dependence, from expression (8.126), the di

ff

ff erential increment in viscosity is

proportional to

ρ T m exp D P

dP ,

D

(8.127)

ρ T m

with D a pressure-dependent parameter, allowing for the pressure dependence of
the activation volume, and dP the di erential increment in pressure. The integral of

(8.127) over pressure is easily converted to an integral over radius r since dP / dr

ff

=
− ρg, where g is the gravitational acceleration at radius r . The viscosity at radius r
is then

ρ T m exp D P

dP
dr dr ,

+ η b r
b

D

η( r ) = η b (8.128)

ρ T m

with b the radius of the core-mantle boundary and η b 3368 Pa s, the dynamic

viscosity at the top of the core. To perform the integration in (8.128), we require
profiles of pressure, density, melting temperature and pressure gradient. The pres-
sure profile can be found by integrating the product of gravity and density for an
Earth model (we use Cal8). The melting temperatures are found by spline inter-
polation onto the Cal8 radii from those tabulated by Stacey (1992, p. 459). The
required profiles are shown in Table 8.5.

=