All about TPW

The Earth's rotation axis is subject to a broad class of perturbations that extend over timescales ranging from sub-second to billions of years. Perturbations in which the rotation axis remains fixed to the solid body, with the axis' movements evident to an (inertial) observer in space, are termed nutation or (at the longest frequency) precession. Notably, the passing of roles of the North Star from one star to another is due to precession. These motions are driven by external torques (from other bodies in the solar system) acting on the rotational bulge.

Perturbations to the Earth's rotation axis are also characterised by reorientations relative to makings anchored to the solid body, changing the latitude of a fixed surface point. This type of reorientation is a response to changes in mass distribution within or on the surface of the Earth.

In true polar wander (TPW), the rotation axis (red dot above) remains fixed in (inertial) space. TPW would not cause the pole to point towards another North Star. Instead, the planet itself reorients with respect to its rotation axis, so a person standing at an high-latitude location might observe that the North Pole is 'drifting' farther and farther away.

To visualise the basic physics behind TPW (i.e., conservation of angular momentum), imagine a spinning ball. If there arises any misbalance in mass, inside or outside the ball, the heavier portion would tend to be 'thrown' outwards so that it is as far from the rotation axis as possible (i.e., to the equator). To an observer standing on the ball, the rotation pole would appear to be shifting away from the 'loaded' side. On Earth, this load might consist of ice sheets, density variations related to convection in the mantle, or contributions from the ocean and atmosphere. These loading processes could cause reorientations that are either cyclical (e.g. the 14-month Chandler wobble) or secular. The latter type is termed True Polar Wander (TPW).

Spinning Ball Analogy

To visualise the basic physics behind TPW, imagine a spinning ball. If there arises any misbalance in mass, inside or outside the ball, the heavier portion would tend to be 'thrown' outwards so that it is as far from the rotation axis as possible (i.e., to the equator). To an observer standing on the ball, the rotation pole would appear to be shifting away from the 'loaded' side.

Role of Equatorial Bulge

Most planets, like the Earth, has an equatorial bulge. As TPW occurs, this bulge is displaced from the reoriented rotational equator. This causes the bulge to act as an excess mass that will tend to move towards the new equator, opposing the initial TPW. The extent and rate of TPW depend not only on the size of the driving mass load, but also on the size and persistence of the equatorial bulge. If the planet is weak, it would deform/adjust such that a new rotational bulge would form around the new equator. If, on the other hand, the planet is stiff in some way, the old bulge would persist and tend towards halting TPW. (See below for the effects of an elastic lithosphere.)

Moreover, planetary equatorial bulges often has additional components generated by processes other than self rotation (e.g. internal convection). Such components would continue to affect TPW for as long as those generating processes persist, in addition to the deformation time required to 'erase' their remnant shape.

While an excess mass (white blob) would cause TPW, the resultant displaced bulge (around the pre-TPW equator marked by the dashed line) would resist it. The rate and extent of TPW depend on the interplay between these driving and resisting forces, which, in turn, depend on various properties of the planet.

Significance of TPW

One application of understanding TPW, apparent from the description above, is to place constraints on internal properties of the Earth. Studies such as those by Steinberger & O'Connell (1997, Nature, 387, 169) and Richards et al. (1997, Science, 275, 372) used TPW as a constraint on, or target for, models of mantle convective flow.

TPW has also been identified as a potentially important control on climate. In ice age studies, observed TPW has been used to estimate the present-day mass balance of polar ice sheets (e.g. Mitrovica et al. 2006, Geophysical Journal International, 161, 491). On the other hand, traditional calculations of ice-age-induced TPW predict an irrecoverable drift of the pole, commonly called 'unidirectional TPW', which has been suggested as a possible mechanism for transitioning out of the ice age cycles (e.g. Sabadini et al. 1982, Nature, 296, 338).

Over longer time scales, TPW with magnitude of tens of degrees over a 10-million-year time scale (as estimated by Besse & Courtillot 2002, Journal of Geophysical Research, 107, 2300) and rapid TPW reaching about 90 degrees within a few million years (e.g. Maloof et al. 2006, Geological Society of America Bulletin, 118, 1099) would also have a significant impact on the Earth's climate.

Why Study TPW?

TPW has been used as either constraints on mantle convective flow, or target results for models of mantle convection.

Ice-age-induced TPW has been suggested as a possible mechanism for transitioning out of ice age cycles.

Rapid and/or large scale TPW suggested by paleomagnetic studies would have significant implications on climate.

Effects of an Elastic Lithosphere

Predictions of rotational stability due to the ice ages or convective motions in the mantle are ultimately founded upon the physical arguments of Gold (1955, Nature, 175, 526). Gold's central argument was that the Earth's rotation axis was inherently unstable over long timescales, since the force resisting TPW would decay to zero as the planet's rotational bulge adjusted to any new orientation (see the section on 'Role of Equatorial Bulge' above). In this view, any load, however small, would eventually reach the rotational equator.

This view has been questioned, in the context of ice-age-induced TPW, by Mitrovica et al. (2005, Geophysical Journal International, 161, 491), who have shown that a more accurate treatment of the background rotational form — one that prevents perfect reorientation — implies a significantly more stable rotation pole than is predicted using the traditional theory.

Their argument is two-parted. First, the Earth's lithosphere has significant elastic strength on ice age time scales. Even though this elastic shell is broken into plates, Chan et al. (2011, Geophysical Journal International, 187, 1319-1333) showed that it still retains elastic strength (in the degree 2 spherical harmonics) equivalent to a uniform, un-broken lithosphere a few tens of kilometres thick (instead of zero thickness).

Moreover, the Earth's equatorial flattening is observed to be flatter than if a spherical Earth was recently 'spun up' to the current rotation rate. This shape is associated with the so-called 'hydrostatic equilibrium'. In other words, the Earth has a shape of a rotating ball of fluid. This might arise in two ways: (i) the lithosphere of the Earth cooled out of an initially molten planet during its formation, or (ii) the current state of rotation has persisted for sufficiently long that all layers of the Earth effectively behave as fluids (i.e., all viscous stresses have relaxed). In both cases, it is important to note that the lithosphere carrying the hydrostatic form is unstressed.

Any TPW-induced adjustments of the equatorial bulge towards the new rotation axis will introduce elastic stresses in the lithosphere. Since this reorientation is acting on an Earth that has elastic strength (as opposed to a perfect ball of fluid), the new bulge will not take on a hydrostatic form, instead leaving behind a remnant component — the so-called 'remnant rotational bulge' (see Willemann 1984, Icarus, 60, 701-709; and Matsuyama 2006, Journal of Geophysical Research, 111, E02003). Simply put, the bulge readjustment caused by TPW will be imperfect, and will retain 'memory' of the initial state of rotation.

The stressing of the lithosphere is akin to the stressing of a rubber shell: the more one tries to deform it, the stronger the restoring force. Given the resistive effect a bulge has on TPW, the remnant bulge's restoring force could significantly slow (or halt, if the restoring force balances the driving force) TPW. This not only stabilises the rotation pole in terms of speed of reorientation, but also the final position, since the final latitude of the excess mass load is determined by the balance between the induced driving force and the restoring force provided by the elastic lithosphere.

Stabilising the Unstable

Traditional theory:

Force resisting TPW would decay to zero as the planet's rotational bulge adjust to any new orientation.

The transience of the bulge resistance means that any load, however small, would eventually reach the equator.

Earth's rotation is inherently unstable.

New theory:

Earth's shape is observed to be shaped more like a rotating ball of fluid, which has a greater flattening in the bulge than if a spherical version of Earth was recently 'spun up' to the current rotation rate.

Lithosphere carries this fluid-like shape/flattening (so-called 'hydrostatic form'), and is unstressed prior to any TPW.

TPW would introduce elastic stress in the otherwise unstressed lithosphere, the physics of which is somewhat similar to deforming an elastic rubber shell.

Combination of the greater flattening and the elastic lithosphere would prevent perfect readjustment of the bulge around the new rotation pole, leaving a 'remnant bulge'.

TPW is greatly slowed and will likely be halted before the excess mass loading reaches the equator.

Earth's rotation is greatly stabilised.

Time Scale and Other Processes

The stabilisation discussed above is not permanent; after all, the Earth's lithosphere is not a perfect rubber shell. There is a time scale beyond which the lithosphere will start behaving mainly viscously instead of mainly elastically. Depending on the exact properties of the lithosphere, this time scale may be as long as millions of years. If the load driving the TPW persisted for shorter than that, the elastic restoring force would ensure the return of the system to its initial rotational state.

In reality, the Earth's current equatorial bulge is even flatter than hydrostatic, and thus providing additional resistance to TPW. This so-called 'excess ellipticity' is associated with convection in the mantle. The persistence of this excess flattening is controlled by the persistence of the convective process(es) driving it. Therefore, it could potentially stabilise the Earth's rotation beyond the time scale after which viscous deformation of the lithosphere becomes significant.

Convection in the mantle is currently adding a greater flattening to the Earth's equatorial bulge. This provides an even greater resistance to TPW than just the previously discussed mechanisms. Depending on the persistence of the convective process(es) sustaining this excess ellipticity, the Earth's rotation pole could be stabilised for a significantly longer period of time (see main text on the left for details).