The dynamics of the Earth's inner and outer cores

Abstract

A brief historical introduction to the study of the Earth's core is followed by a review of recent advances in the dynamics of the liquid outer core and the rotation of the solid inner body. In particular, the scaling of the fluid motion equations is reviewed and a detailed derivation of the 'subseismic equation' which governs small-amplitude, low-frequency oscillations is given following methods first outlined by Rochester. Attention is given to the compressibility of the outer core and its role in dynamo theory as well as its effect on Proudman-Taylor flow, where it produces a fluid motion with helicity. The theorem of J B Taylor is also generalised to real core compressibility. Rotational motions of the inner core are examined under the strong gravity torque exerted on it by the rest of the Earth. Both the motion ignoring back reaction on the mantle's rotation and the full coupled problem are solved. The gravity torque is shown to be five orders of magnitude larger than inertial coupling which was previously considered dominant. The 'rapid' regular precessional mode which is nearly diurnal in space, largely ignored in the modern literature, may play a role in the time evolution of the dipole part of the main magnetic field. Inner-core behaviour during field reversals remains to be investigated. Energy considerations are shown to require nearly equilibrated electromagnetic torques on the inner core and mantle if they are to have differential rotations with respect to the outer core. For steady dynamos they must vanish identically. A variational principle for the induction equation is established which allows the constraints of vanishing torques to be introduced via Lagrange multipliers. It will also permit the abandonment of the traditional spherical vector harmonics as approximating functions, which are notoriously poorly convergent, in favour of finite elements. It is anticipated that torque constraints will limit the size of toroidal fields in the outer core but re-solution is required.

Possible relations between the rotational axis of the Earth's inner core and the magnetic dipole axis

“…Smylie et al (1984) and Smylie (1984, 1989). Assuming that the dipole field is frozen within the inner core, they suggested that the relative retrograde precession of the inner core resulting from their model can possibly be detected by a corresponding relative precession of the geomagnetic dipole axis.…”

Core Oscillations and Their Detection in Superconducting Gravimeter Records

“…Detailed scaling arguments (SMYLIE and ROCHESTER, 1981;SMYLIE et al, 1984), show that p1 is of order FRC where FR is the Froude to Rossby number ratio Il2L/go, and C is the compressibility number pogoL/A with go, po and A indicating characteristic values of gravity, density and adiabatic bulk modulus, respectively. Substitution of values appropriate to the Earth's core indicates that FRC can amount to no more than a few parts in a thousand, and that even in the subtidal band neglected acoustic effects give corrections to the displacements in the range of one percent.…”

“…For the outer core, acoustic periods can be no more than the order of 10 minutes, and even in the subtidal band acoustic effects amount to corrections of only a few per cent. At longer periods, the dominant Coriolis acceleration gives rise to a small flow pressure effect on the density measured by the product of the Froude to Rossby number ratio, FR, with the Compressibility number C (SMYLIE et al, 1984). FRC is at most only a few parts in a thousand and is independent of the period.…”

The Gravity Effect of Core Modes for a Rotating Earth

“…(25)) which include the inertia, pressure, buoyancy and Coriolis forces for infinitesimal oscillations about hydrostatic equilibrium. The general problem is outlined in CROSSLEY (1984) and SMYLIE et al (1984). In particular we use the following expansion of the Lagrangian displacement where the degree 1 is incremented by 2.…”