A core that wobbles at its own period

The Earth’s axis does not simply precess. The circle it traces around the pole of the ecliptic every 25,800 years is the average of a motion with ripples in it: nutations, forced by the Sun and the Moon at periods set by their orbits — 18.6 years from the regression of the Moon’s nodes, a year and half a year from the Sun, a fortnight and half a month from the Moon. For a rigid Earth the amplitudes of those ripples follow from the planet’s moments of inertia and the orbits of the two bodies, and the calculation was done to high precision long before anyone could measure the motion well enough to test it.

The Earth is not rigid. Its mantle is solid rock, but beneath 2,900 kilometres of it lies a liquid iron core, and a liquid core in a flattened cavity is not simply carried round with the shell. It has a mode of motion of its own: its rotation axis can tilt slightly away from the mantle’s, and the pressure of the fluid against the flattened boundary pushes it back. That is a free oscillation, and like the Chandler wobble of the whole planet, it has its own period. Any forced motion near that period is resonantly amplified.

Two frames, one mode #

The free mode has two descriptions, according to the frame. Seen from the rotating Earth, the core’s axis is offset from the mantle’s and circles it, in the direction opposite to the Earth’s rotation, a little faster than once a day: the nearly diurnal free wobble, at about 1.0023 cycles per sidereal day. Seen from space, the same motion is a slow retrograde nodding of the Earth’s axis — the free core nutation — with a period of about 430 days. The two frequencies differ by exactly one cycle per sidereal day, because the rotating frame itself turns once per sidereal day.

A free mode of this kind needs two ingredients: something that can move independently — here the fluid core, which slides past the mantle almost without friction — and something that pushes it back when it does. The period comes from the physics of a spinning fluid in a flattened container. If the core’s boundary were a perfect sphere, a tilt of the core’s rotation axis would meet no resistance, and the core could turn about any axis at no cost. The boundary is flattened by the Earth’s rotation, by about one part in four hundred, and a fluid spinning about an axis tilted to the symmetry axis of an ellipsoidal cavity presses harder on some parts of the wall than others. The torque between the fluid and the wall is proportional to the flattening, and it drives the core’s axis round in a slow precession. The frequency is, to good accuracy,

$$\sigma_{\rm FCN} \approx -\frac{A}{A_m}\,(e_f - \beta),$$

in cycles per sidereal day, where $A/A_m \approx 1.13$ is the ratio of the whole Earth’s moment of inertia to the mantle’s, $e_f$ the dynamical flattening of the core–mantle boundary, and $\beta$ a correction for the boundary’s elastic yielding under the fluid’s pressure. A flattening of two parts in a thousand gives a period of about 430 days.

Where the resonance falls #

The Sun and the Moon drive the Earth’s axis at a handful of frequencies, each of which appears in space as a nutation term, prograde or retrograde, with an amplitude set by the orbital geometry. Part of the Earth’s response to each involves the core, and that part is multiplied by a resonance factor proportional to $\sigma_{\rm FCN}/(\sigma - \sigma_{\rm FCN})$. The factor is largest for the forced term nearest the free one.

That term is the retrograde annual nutation. The Sun’s annual motion drives the axis at a period of exactly a year in both directions, and the retrograde year, 365 days, lies within a sixth of the core’s 430. The resonance factor for it is about five and a half, against about one for the 18.6-year terms, a half for the prograde annual term and less for the rest. The retrograde annual nutation is therefore the one whose observed amplitude departs most from the rigid-Earth prediction, and the departure is the core showing itself.

That departure was one of the first things very long baseline interferometry measured when it began to track the orientation of the Earth against distant quasars in the 1980s. An angle measured against a quasar is referred to objects so far away that they define an inertial frame, and the Earth’s axis could be followed against them to a fraction of a milliarcsecond. The retrograde annual nutation came out different from the theory of an Earth in hydrostatic equilibrium by about a milliarcsecond — small, and unambiguous. It was not the only discrepancy — the 18.6-year and semiannual terms also differed from theory by smaller amounts — but the pattern of discrepancies across the terms, largest near 430 days retrograde and falling off away from it, was the pattern of a resonance, and it placed the resonance where the gravimeters had already put it.

Why one term pins the period #

The annual term’s nearness to the resonance is what makes it a measurement rather than merely a correction. Move the free core nutation from its observed 430 days to the 460 a hydrostatic Earth would have, and the resonance factor for every term changes — but for most of them hardly at all, and for the retrograde annual term a great deal.

That sensitivity is the lever. The amplitude of the retrograde annual nutation, measured against quasars to a few microarcseconds, fixes where the resonance must be; the other terms, which barely respond, fix the coefficients that multiply the factor. Fitting both together gives the free core nutation’s period to a fraction of a per cent without ever observing the free oscillation directly — the forced terms alone locate the resonance, in the way that a tuning fork’s frequency can be found from how strongly it responds to notes played near it.

The same mode in the daily tides #

The mode that distorts the annual nutation in space also distorts the tides on the ground, and the reason is the frame relation. A torque that makes the axis nutate with frequency σ in space appears on the rotating Earth at 1 − σ cycles per sidereal day. The Sun and Moon’s pull on the equatorial bulge that drives nutation is, seen from the ground, the diurnal tide: the daily rise and fall of the solid Earth and the sea, with lines named O1, P1, K1, ψ1 and φ1.

The map is exact, and it means the resonance has to appear among the tides too.

That is how the resonance was first detected, before interferometry could see it: gravimeters recording the tidal variations of gravity at a single site found the Earth’s response to the diurnal lines varied with frequency in a way that only a resonance just above once a day could explain, and fitting it gave a period for the free wobble. The ψ1 tide is tiny — a small line in the solar tide, a thousand times weaker than K1 — but its closeness to the resonance makes its response a sensitive probe, and superconducting gravimeters running for years at quiet sites measure it well enough to pin the resonance frequency to a few per cent. Two measurements that share nothing — the orientation of the axis against quasars, and the rise and fall of the ground under a gravimeter — find the same period.

That kind of agreement is worth dwelling on. The quasar measurement is purely kinematic, a direction in space; the gravimeter measurement is a local change of gravity, a response of an elastic planet to a tide, with its own corrections for the ocean and the atmosphere. Each is a different window onto the same eigenmode, and the eigenmode’s frequency is a property of the core’s shape that neither measurement touches directly.

A period that measures a shape #

The free core nutation’s period is, through the formula above, a measurement of the flattening of the core–mantle boundary. The prediction for a boundary in hydrostatic equilibrium — shaped only by the Earth’s rotation and the density layering of the interior, the same structure that sets the moment of inertia — is about 460 days. The observed period is 430.

A few per cent sounds small, and in terms of shape it is. Translated into relief, it is several hundred metres.

What makes the boundary more flattened than hydrostatic is not settled. One possibility is real topography on the boundary, maintained by the flow of the mantle above it: the dense, sinking slabs of old ocean floor and the hot, rising regions beneath Africa and the Pacific push and pull on the boundary, and seismic studies find evidence of relief of a few kilometres at shorter wavelengths. Seismology sees the boundary through waves that reflect from it or pass through it, and its maps are sharpest at the scales of hundreds of kilometres; the nutation sees only the single, longest-wavelength component of the shape that exerts a torque, and sees it with far better precision. The two techniques overlap little, and a model of the boundary that satisfies both is still being built. Another is a density anomaly just above or below the boundary with the same effect on the moments of inertia. A third involves the magnetic field: the field lines threading the boundary couple the core to the mantle electromagnetically, adding a restoring torque that mimics extra flattening. Each explanation predicts a slightly different damping of the mode as well as its period, and the observed damping — the free core nutation has a quality factor of about twenty thousand — is one of the constraints that distinguishes them.

The free mode itself #

The resonance is only half of the signal. The free core nutation is also excited as a free oscillation, by the atmosphere and oceans pushing on the mantle at nearly the right frequency, and it appears in the interferometric record as a nodding of the axis with an amplitude of a hundred to a few hundred microarcseconds, a period near 430 days and an amplitude and phase that drift over years. That drift is the signature of random excitation: a lightly damped oscillator kicked irregularly by weather keeps ringing at its own frequency with an amplitude that wanders.

That behaviour is the same as the Chandler wobble’s, and for the same reason. The Chandler wobble is a free mode of the whole Earth, ringing at about 433 days, damped and re-excited by the atmosphere and oceans. The free core nutation is a free mode of the core relative to the mantle, ringing at 430 days in space — a coincidence of periods that is only a coincidence, since the two depend on different flattenings, and the two motions are distinguished by the frame in which each has its period. The Chandler wobble is a motion of the axis relative to the Earth’s crust; the free core nutation a motion of the axis relative to the stars. Neither can be predicted beyond a few years, because the weather that excites them cannot. That has a practical consequence for anyone who needs the Earth’s orientation in advance — spacecraft navigators, satellite geodesists — because the free core nutation’s contribution, a few hundred microarcseconds, is larger than the accuracy they need. It is modelled empirically, from the most recent interferometric sessions, and extrapolated a few months ahead with an uncertainty that grows with time, in the same way that the Chandler wobble’s contribution to the position of the pole is forecast.

The core and the length of the day #

The free core nutation is one of two ways the core’s motion shows up at the surface. The other is the length of the day. The Earth’s rotation rate varies by a few milliseconds over decades, far more than the atmosphere and oceans can account for, and the residual is angular momentum exchanged between the mantle and the liquid core along the rotation axis — the core speeding up while the mantle slows, and the reverse. The nutation is the same coupling in the perpendicular direction: the core’s axis tilting relative to the mantle’s rather than spinning faster or slower.

Both couplings pass through the same boundary and are made of the same ingredients — pressure on its topography, magnetic stresses in the field threading it, and viscous friction in a thin layer at its base — and a model of the boundary has to explain both. The nutation measures the flattening, the part of the boundary’s shape that exerts a torque about an equatorial axis; the decade variations in the day measure the coupling about the polar axis. Together they are two components of the torque between the Earth’s two largest reservoirs of angular momentum. The nutation is driven from outside, by the Sun’s and Moon’s pull on the equatorial bulge — a tidal force, a difference of gravity across the planet — and the core’s resonance is how the planet’s inside answers it.

What the pictures simplify #

The resonance factor drawn is the shape of the resonance with its amplitude left out: the actual change to each nutation is the factor times a coefficient that depends on the core’s moment of inertia, its coupling to the mantle and the elastic response of the whole planet, which the published nutation theories fit to the data. The frequencies are real; the heights are relative.

The theory used in practice, adopted by the International Astronomical Union in 2000, contains not one resonance but several: besides the free core nutation, the Chandler wobble, a prograde free mode of the solid inner core, and a free mode of the inner core’s tilt. The inner-core modes have long periods and small effects, and one of them has been detected only tentatively. The fits that pin the free core nutation’s period to half a per cent use the full set, with electromagnetic and viscous coupling at both boundaries of the fluid core.

And the formula for the period is the leading-order version. The full expression includes the elastic deformability of the mantle, the ocean’s response and the coupling torques, and those change the period by a few per cent — which is the size of the effect being measured. The hydrostatic prediction of about 460 days is the result of that fuller calculation for a hydrostatic Earth, not of the formula alone, and the figure uses the formula only to convert the difference between prediction and observation into a flattening.

The planet’s interior from its orientation #

The general point is that the orientation of a planet in space is an instrument for its interior. Mercury’s libration weighs its core by showing that only part of the planet follows its mantle; the Earth’s nutations weigh the flattening of its core by showing a resonance at a frequency set by that flattening. In both cases what is measured is a direction, and what is inferred is the shape and state of something no instrument will reach. For the Earth the direction is measured to tens of microarcseconds — the angle a coin subtends from across a continent — and the inference reaches the bottom of the mantle.

Still open: what holds the core’s boundary out of shape #

The excess flattening, a few hundred metres of equivalent relief, is established; its cause is not. Mantle flow, a density anomaly and electromagnetic coupling each account for it with plausible parameters, and they predict different things about the damping of the free core nutation, about the variations in the length of day on decade timescales that also involve the core, and about the long-wavelength shape of the boundary that seismology is slowly mapping. The electromagnetic coupling in particular would require a magnetic field at the top of the core several times stronger than the part of the field that reaches the surface — a field that cannot be observed from above because the mantle screens its small-scale structure. If the nutation data require it, the Earth’s orientation in space will have measured the strength of a magnetic field that no magnetometer can see.