  The diurnal tidal forces can excite a normal mode of the Earth's core, the
free inner core nutation (FICN), which is characterized by a tilt of the
rotation axis of the inner core with respect to the rotation axis of the outer
core. The differential rotation between the inner core and the outer core
induces fluid motions in the outer core and gives rise to Ohmic dissipation in
the presence of the Earth's internal magnetic field. Nutation measurements can
reflect such dissipation if it is sufficiently strong and thus can provide
insights into the properties and dynamics of the Earth's core. In this study we
perform a set of numerical calculations of the linear perturbations in the
outer core induced by the FICN at very low Ekman numbers (as small as
$10^{-11}$). Our numerical results show that the back-reaction of the magnetic
field notably alters the structure and length scale of the perturbations
induced by the FICN, and thus influences the Ohmic dissipation resulting from
the perturbations. When the Ekman number is sufficiently small, Ohmic
dissipation tends to be insensitive to the fluid viscosity and to the magnetic
diffusivity, which allows us to estimate the Ohmic dissipation associated with
the FICN without relying on an extrapolation. In contrast to the results of
Buffett (2010b), the estimated Ohmic dissipation based on our numerical
calculations is too weak to account for the observed damping of the FICN mode.
This also implies that nutation measurements cannot provide effective
constraints on the strength of the magnetic field inside the Earth's outer
core.
