The mantle-inner core gravitational mode of oscillation in a strong magnetic field regime

TL;DR

The paper addresses whether the mantle-inner core gravitational (MICG) mode remains an independent normal mode in a strongly magnetized Earth-like core. It develops a 1D cylindrically layered core model to couple MICG with torsional oscillations (TO) via Alfvén waves launched by the tangent cylinder, solving an eigenvalue problem for complex frequencies with EM and viscous diffusion. The key finding is that MICG exists as a distinct mode only if Alfvén waves are attenuated before crossing the core width; with a few mT internal field, Alfvén waves traverse the core and MICG is absorbed into the TO spectrum, though the MICG period still resonates TO modes and marks a transition in their angular-momentum budgets. The result implies that the observed ~6-year length-of-day signal cannot be MICG and is instead linked to TO dynamics or Alfvén wave propagation, constraining mantle conductance and core–mantle coupling, and guiding interpretation of subdecadal inner-core rotations.

Abstract

The mantle-inner core gravitational (MICG) mode is the free mode axial oscillation between the mantle and inner core sustained by the gravitational torque between their degree 2 order 2 density structures. Here, we investigate how the MICG mode is affected by oscillations of cylindrical surfaces in the fluid outer core in the form of Alfvén waves. The latter are triggered by oscillations of the tangent cylinder (TC) moving jointly with the inner core and propagate away from the rotation axis. We show that the MICG mode remains a distinct normal mode of oscillation of the core-mantle system only when the triggered Alfvén waves are attenuated before they traverse the width of the fluid core. For an internal magnetic field strength of a few mT, as we expect in Earth's core, Alfvén waves can readily traverse the width of the core, and the MICG mode is absorbed into the spectrum of torsional oscillation (TO) modes. The MICG period retains a dynamical influence, acting as a point of resonance for TO modes, and marking the transition from a TO mode in which the motion of the TC (including the inner core) is weakly impacted by gravitational coupling to one in which the oscillating motion of the TC is strongly restricted. Our results imply that the observed 6-year periodic signal in the length of day cannot be interpreted as the signature of the MICG mode and must instead be caused by TO modes, or more generally, by the propagation of Alfvén waves.

Figures (4)

• Figure 1: Frequency and $Q$ of a subset of the normal modes obtained with $\overline{\Gamma}=3 \times10^{20}$ N m, (a) $\overline{B}_s=0.1$, (b) $\overline{B}_s=0.15$, and (c) $\overline{B}_s=0.3$, and different choices of mantle conductance $G_m$ (see legend). Black squares show the TO modes in the absence of gravitational coupling: $\overline{\Gamma}=0$. In each panel, the predicted $\omega_{micg}$ from Eq. \ref{['eq:micg2']} is indicated by the pink vertical line. The angular velocity structure of the modes identified by a purple outline is shown in Figure \ref{['fig:omegaf']}.
• Figure 2: The angular velocity of the inner core $\Omega_i$ (thick light blue), mantle $\Omega_m \times 100$ (thick orange) and of the fluid core $\Omega_f$ (thin dark blue) as a function of cylindrical radius ($s/r_f$) for the modes identified by a purple outline on the respective panels of Figure \ref{['fig:wQ']}. The dashed black line corresponds to the prediction $\Omega_m = - (C_{tc}/C_m) \Omega_i$ (multiplied by 100) when the exchange of angular momentum is between the mantle and whole of TC. In each panel, the phase is chosen when $\Omega_i$ is maximum in the prograde direction and the amplitude is normalized such that $\max | \Omega_f |=1$.
• Figure 3: The period (a) and $Q$ (b) of the first four TO harmonics (coloured lines, legend) as a function of $\overline{\Gamma}$. The black line in (a) is the period of the MICG mode predicted from Eq. \ref{['eq:micg2']}. In (b), results are shown for $\tau=100$ years (solid lines), $\tau=10$ years (thin solid lines) and $\tau=3$ years (dashed lines).
• Figure 4: The angular velocity as a function of cylindrical radius ($s/r_f$) of the (a, d) first, (b, e) second, and (c, f) third TO harmonics for $\overline{\Gamma}=10^{19}$ N m (left column) and $\overline{\Gamma}=10^{22}$ N m (right column). Thin blue lines: $\Omega_f$, thick light blue lines: $\Omega_i$ and thick orange lines: $\Omega_m \times 100$. The dashed black line corresponds to the prediction $\Omega_m = - (C_{tc}/C_m) \Omega_i$ (multiplied by 100). The dashed red line corresponds to the prediction of $\Omega_m = - L_{rotc}/C_m$ (multiplied by 100), where $L_{rotc}$ is the net angular momentum of the ROTC. In each panel, the phase is chosen when $\Omega_i$ is maximum in the prograde direction and the amplitude is normalized such that $\max | \Omega_f |=1$.