The core-mantle mode of gravitational oscillation

We present the characteristics of a mode of axial oscillation between Earth's mantle, fluid core and solid inner core that has not been previously reported. The mode involves a quasi-rigid rotation of the fluid core outside the tangent cylinder (TC) exchanging its angular momentum with the mantle via a three step process. First, by a magnetic torque with the fluid inside the TC; second by a magnetic torque between the latter and the inner core; and finally by a gravitational torque between the inner core and mantle. Although the gravitational torque is purely between the mantle and inner core, the mode involves an oscillation of the whole of the core, and we refer to it as the core-mantle gravitational (CMG) mode. This form of gravitational oscillation occurs when the magnetic field within the core is sufficiently strong that the propagation time of Alfvén waves is shorter than the mode period, which is the case for Earth. We show how the period, quality factor $Q$ and structure of the CMG mode depends on the strength of the gravitational torque and viscous relaxation time $\tau_i$ of the inner core. For Earth, the CGM mode period should be in the range of 40 to 100 years, but viscous relaxation of the inner core likely implies a small $Q$, below 1 if $\tau_i<10$ years. Our results suggest that the CMG mode may act to amplify resonantly, though only modestly, multi-decadal changes in the length of day driven by zonal accelerations in the fluid core.