3.1 Euler Poles & Plate Rotation
Euler's Rotation Theorem
Euler's rotation theorem (1776) states that any displacement of a rigid body on the surface of a sphere can be described as a single rotation about an axis passing through the center of the sphere. This axis intersects the surface at two points: the Euler pole and its antipode. For plate tectonics, this means the relative motion between any two plates is fully specified by three parameters: the latitude and longitude of the Euler pole, and the angular rate of rotation about that pole.
The theorem has profound implications. All points on a plate trace small circles around the Euler pole during motion. Transform faults, which accommodate lateral plate sliding, lie on these small circles and thus point directly toward the Euler pole. Mid-ocean ridge segments, in contrast, are perpendicular to the spreading direction and lie on great circles through the pole.
Fundamental Velocity Equation
\[ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{r} \]
where v is the linear velocity at a point on the plate surface, ω is the angular velocity vector (pointing along the rotation axis), and r is the position vector from Earth's center to the surface point.
Surface Speed
\[ |v| = |\omega| \, R \, \sin(\Delta) \]
where R is Earth's radius (~6371 km), and Δ is the angular distance from the Euler pole to the point of interest. Velocity is zero at the Euler pole itself (Δ = 0) and maximum 90° away.
Angular Velocity Vector
The angular velocity vector ω fully encodes the pole location and rate of rotation. In a geocentric Cartesian coordinate system (x-axis through 0°N/0°E, y-axis through 0°N/90°E, z-axis through the North Pole), the vector is expressed as:
Cartesian Components of ω
\[ \boldsymbol{\omega} = (\omega_x, \, \omega_y, \, \omega_z) \]
\[ \omega_x = |\omega| \cos\Phi \cos\Lambda \]
\[ \omega_y = |\omega| \cos\Phi \sin\Lambda \]
\[ \omega_z = |\omega| \sin\Phi \]
where Φ is the latitude and Λ is the longitude of the Euler pole, and |ω| is the magnitude (angular speed) in °/Ma.
The Cartesian representation is convenient for algebraic operations: to obtain the relative angular velocity of plate A with respect to plate C from known pairs, we simply add vectors: ωAC = ωAB + ωBC. This additivity holds exactly because angular velocity vectors combine linearly for relative rotations at a given instant.
Reference Frames for Plate Motions
Plate tectonics inherently describes relative motions between plates. There is no absolute rest frame anchored to the deep mantle that can be measured directly. However, two principal reference frames are used to approximate “absolute” plate velocities.
Hotspot Reference Frame
Assumes that volcanic hotspots (e.g., Hawaii, Iceland, Reunion) are anchored to relatively stationary plume sources in the deep mantle. By tracking the age progression of hotspot chains on a plate, one infers the plate's absolute velocity. The HS3-MORVEL56 model (Gripp & Gordon, 2002; updated) is the most widely used variant. Limitations include evidence that plumes themselves drift at ~1–2 cm/yr, introducing systematic errors.
No-Net-Rotation (NNR) Frame
Defined so that the integrated angular momentum of all lithospheric plates equals zero. This is a mathematical construction: it distributes net rotation equally, so no single plate is “preferred.” The NNR-MORVEL56 model is the IERS standard for geodetic applications. GPS velocities are reported in this frame (ITRF). In this frame, the Pacific plate moves NW at ~6.7 cm/yr, while in the hotspot frame it moves at ~10 cm/yr.
The difference between frames reflects the net lithospheric rotation — the bulk westward drift of the lithosphere relative to the underlying mantle. This signal is ~0.44°/Ma about a pole near (56°S, 70°E) in the HS3 model, and its physical origin (lateral mantle flow vs. tidal drag) remains debated.
Major Plate Angular Velocities (MORVEL)
The MORVEL model (DeMets et al., 2010) provides the current best estimate of angular velocities for 25 tectonic plates relative to the Pacific plate. Selected values relative to the NNR reference frame are given below.
| Plate Pair | Pole Lat (°N) | Pole Lon (°E) | |ω| (°/Ma) |
|---|---|---|---|
| Pacific–North America | 50.0 | −78.2 | 0.755 |
| Africa–North America | 80.4 | 56.2 | 0.199 |
| Africa–South America | 62.5 | −39.4 | 0.326 |
| Africa–Eurasia | 21.0 | −20.6 | 0.060 |
| India–Eurasia | 28.6 | 11.4 | 0.544 |
| Pacific–Antarctic | 64.3 | −83.7 | 0.870 |
| Nazca–Pacific | 55.6 | −90.1 | 1.362 |
| Nazca–South America | 56.0 | −94.0 | 0.720 |
Source: DeMets, C., Gordon, R.G., & Argus, D.F. (2010). Geologically current plate motions. Geophysical Journal International, 181, 1–80.
Key Concepts
Rigid Plate Assumption
Euler's theorem requires plates to behave as rigid caps on a sphere. Deformation is concentrated at narrow plate boundaries. The rigid-plate approximation holds to within ~2 mm/yr for most major plate interiors, validated by GPS.
Transform Fault Geometry
Transform faults trace small circles about the Euler pole. The first determination of a pole location (Morgan, 1968) used the azimuths of Atlantic transforms to locate the Africa–North America pole.
Velocity Varies with Distance
Because v = |ω|R sin(Δ), spreading rate at a mid-ocean ridge increases with distance from the Euler pole. The South Atlantic spreading rate increases from ~2 cm/yr near the pole to ~3.5 cm/yr in the central Atlantic.
Relative vs. Absolute
Relative motions are directly measurable; absolute motions require a reference frame. Plate circuit closure ensures consistency: ωAC = ωAB + ωBC. Independent plate pairs serve as validation checks.