Euler Poles

The earth has 7 major plates: North America, South America, Antarctica, Africa, Eurasia, Australia, and Pacific.  All but the Pacific plate include a major continent, and even the Pacific plate includes a sliver of California west of the San Andreas fault.  There are another 8 smaller plates: India, Arabia, Caribbean, Scotia, Cocos, Nazca, Juan de Fuca, and Philippine.  Some additional plates include Somalia, and breaking the Juan de Fuca plate into three parts, which brings the total to about 40-50 plates. As with many things in science, there are splitters and lumpers.  Lumpers look at the big picture, and if several small plates have almost identical motions, the lumpers treat them as a single plate.  When splitters see different motions, they create new plates, but may not have the data to get accurate motions for each.  There are also some boundaries which are very diffuse, and spread over a large area.

The 25 plates in the MORVEL (mid ocean ridge velocity, Demets and others, 2010) model.  Red cross hatched regions have problematic boundaries. NNR-MORVEL56 (Argus and others, 2011) has 56 plates; the 31 additional plates cover 2.8% of the earth's surface area.

The motion of the plates is relative, since there is no fixed spot of the earth.  Other relative motions in geology  include sea level rise (the same result would occur if the land were falling, and in general both land and sea level are changing), and the motion of faults. For plates we can hold one plate fixed, and calculate the motion of the other plates with respect to it.  Alternatively, Jason Morgan proposed that hot spots deep within the mantle provide a fixed reference frame, and that we can calculate the motions of the plates with respect to the hot spots, after removing some hot spots that do move.

We can measure current plate motions using long term GPS monitoring, or very long baseline interferometery (VLBI), which uses radio telescopes to measure the changing time differences in the arrival of random noise at widely separated observatories.  Motions are measured in 10's of mm per year, the preferred unit since it is the same as km per million years.

The motion of a plate on the spherical surface of the earth can be modeled as rotation about an Euler pole. This is named for the Swiss mathematician and physicist Euler (1707-1783), who developed the mathematics involved in 1776 without any appreciation of its application to the earth which would not come until the 1960's.

The Euler pole consists of the location of the pole of rotation, and an angular rotation rate usually given in degrees per million years.  The Euler pole is not related to either the Earth's pole of rotation, or the magnetic pole.  Like the rotational or magnetic pole, the rotation axis intersects the earth's surface in two places.  Some authors (like Cox and Hart ) prefer to select the pole that provides a positive rotation angle with the right hand rule; others prefer the pole on the opposite side of the earth with a negative rotations (such as Duncan and Richards,  but they are not always consistent).

At a pole of rotation the tangential velocity will be zero, and it will increase to a maximum 90 degrees away from the pole.  This could be regarded as an Euler equator, and the increase goes with the cosine of the Euler latitude.

Rotation of North America about its Euler pole in the APKIM8.80 model. The pole, shown by a green cross, is west of Ecuador and does not lie within the North American plate, which is typical for an Euler pole. The tangential velocities increase moving away from the Euler pole.

Along a ridge, the ridge segments will point toward the Euler pole and will lie on great circles through the Euler pole.  The transform faults separating the ridge segments will lie on small circles.  Some depictions of plate boundaries lump the transforms with the ridges, because geometry on a sphere will not allow long straight ridges, while other depictions preserve the distinction and show alternating ridge and transform segments.  We will see that each segment has distinctive fault and earthquake types.

There are three types of poles:

• Current instantaneous poles: the motions of the plates at the present, measured very accurately by GPS or VLBI.
• Total reconstruction poles: this takes a plate from its current location back to its location as a particular point in the past.  This will almost certainly not be the actual path followed, because as soon as there are three plates, the rotation poles for at least two of them must migrate over time.  The correct path would feature a series of rotations about different stage poles.  MICRODEM includes models from Cox and Hart and Duncan and Richards
• Stage poles : the pole that describes the motion during a particular time interval. The three plate problem in plate tectonics says that if you consider any three plates, the poles between them cannot remain constant.  Thus if you want to reconstruct past motions, you need a series of stage poles, each valid only for a certain time period.

The three plate problem in plate tectonics says that with three plates, the poles to at least two must change with time, and probably all three will. The map to the left shows the total reconstruction poles for North American from Cox and Hart.  Note the steady migration of the poles.  A similar pattern would hold for the stage poles which governed the motion during each time interval. Each Euler pole intersects the surface at two locations; since these are all in the Northern Hemisphere, the other will be in the Southern Hemisphere and actually closer to North America.

The motion of a single plate, in any reference frame, does not prove to be particularly interesting.  The relative motion between two plates is much more interesting, or the motions of a triple junction.

Last revision 9/19/2022