3.4 Plate Reconstructions

Finite Rotations

A finite rotation describes the total displacement of a plate from some time t in the past to the present day. It is specified by a rotation pole (latitude, longitude) and a rotation angle θ. The rotation operator R(θ, pole) transforms any point from its past position to its present position on the plate.

Finite rotations are determined by matching conjugate features across plate boundaries: matching magnetic isochrons on opposite sides of a mid-ocean ridge, aligning continental margins during rifting, or fitting fracture zone traces. For oceanic plates, the primary data source is the record of magnetic anomalies correlated with the Geomagnetic Polarity Time Scale (GPTS).

Finite Rotation Matrix

\[ R(\theta, \hat{\mathbf{e}}) = \cos\theta \, \mathbf{I} + (1 - \cos\theta) \, \hat{\mathbf{e}} \hat{\mathbf{e}}^T + \sin\theta \, [\hat{\mathbf{e}}]_\times \]

This is Rodrigues' rotation formula, where I is the 3×3 identity matrix, ê is the unit vector along the rotation axis, and [ê]_× is the skew-symmetric matrix of ê. This produces the 3×3 rotation matrix that transforms Cartesian coordinates from past to present positions.

Stage Poles & Stage Rotations

While finite rotations describe the total displacement since time t, stage rotations describe the motion during a specific time interval [t₁, t₂]. Stage rotations are computed as the difference between two finite rotations and represent the actual plate motion during that interval.

Stage Rotation from Finite Rotations

\[ R_{\text{stage}}(t_1 \to t_2) = R(t_2) \cdot R(t_1)^{-1} \]

where R(t₁) is the finite rotation at time t₁ (older) and R(t₂) at time t₂ (younger). The inverse R(t₁)⁻¹ undoes the older rotation, and the product isolates the motion during the interval. For rotation matrices, the inverse is simply the transpose.

The pole of the stage rotation (the stage pole) is generally different from the finite rotation poles at t₁ or t₂. Stage poles trace out the history of the instantaneous Euler pole through time, revealing when and how plate motion changed direction. Major plate reorganizations (e.g., the ~47 Ma bend in the Hawaiian-Emperor chain) show up as abrupt shifts in stage pole location.

Non-Commutativity of Finite Rotations

A critical mathematical subtlety: finite rotations on a sphere do not commute. That is, performing rotation A followed by rotation B gives a different result than B followed by A.

Non-Commutativity

\[ R_A \cdot R_B \neq R_B \cdot R_A \quad \text{(in general)} \]

This has practical consequences: plate circuits must be composed in the correct temporal order. However, infinitesimal rotations (angular velocity vectors) do commute, which is why the instantaneous plate velocity closure condition ω_AB + ω_BC + ω_CA = 0 uses simple vector addition.

The non-commutativity becomes significant for large rotation angles (>10°) and distant time reconstructions. When composing rotations through a plate circuit (e.g., Africa relative to East Antarctica via India), each step must be applied as a matrix multiplication in the correct sequence: R_total = R_n · R_n−1 · ... · R₁.

Apparent Polar Wander Paths

Apparent polar wander paths (APWPs) are the oldest and most powerful tool for reconstructing plate motions on continents, where oceanic magnetic anomalies are absent. The method is based on paleomagnetism: when rocks form, magnetic minerals record the direction of Earth's magnetic field. If the field is assumed to be a geocentric axial dipole (GAD), the recorded direction gives the latitude and azimuth of the sampling site relative to the magnetic pole.

As a plate moves and rotates, the apparent position of the magnetic pole — computed from the rock's remanence — traces a path in geographic coordinates. This path is not real polar motion but rather the inverse of the plate's own motion. By comparing APWPs of two continents, one can determine their relative motion history.

Key APWP Applications

Reconstructing Pangaea: matching APWPs of Gondwana and Laurasia confirms their assembly ~320 Ma
Tracking India's northward journey: India's APWP shows rapid motion (~18 cm/yr) before the ~50 Ma collision with Eurasia
Detecting true polar wander (TPW): coherent rotation of all APWPs together indicates whole-mantle reorientation
Pre-Pangaean reconstructions: Rodinia (~1.0 Ga) is constrained primarily from paleomagnetic data

Magnetic Isochrons & the Ocean Floor Record

The oceanic crust preserves a continuous record of plate motion through magnetic anomaly lineations (isochrons). As new crust forms at a mid-ocean ridge, it acquires the remanent magnetization of the ambient field. Reversals of the geomagnetic field create alternating stripes of normal and reversed polarity that are symmetric about the ridge axis.

Each anomaly is dated by correlation with the Geomagnetic Polarity Time Scale (GPTS), which is calibrated independently using radiometric dating of volcanic rocks. The current standard GPTS (GTS2020) provides a continuous reversal chronology back to ~170 Ma (the age of the oldest oceanic crust). Key anomalies used for reconstructions include:

Anomaly	Age (Ma)	Chron Name	Significance
1n	0–0.78	Brunhes	Current normal polarity epoch
2A	2.58–3.58	Gauss	NUVEL-1A reference anomaly
5	9.78–11.04	C5n	Key Neogene calibration point
13	33.1–33.7	C13n	Eocene–Oligocene boundary
21	47.3–48.6	C21n	Hawaiian-Emperor bend (~47 Ma)
34	83.6–121	CNS	Cretaceous Normal Superchron (no reversals)
M0–M29	121–157	M-sequence	Jurassic–Early Cretaceous; oldest isochrons

GPlates: Software for Plate Reconstruction

GPlates is an open-source, cross-platform software application for interactive visualization and manipulation of plate tectonic reconstructions. Developed by the EarthByte Group at the University of Sydney in collaboration with Caltech and the Geological Survey of Norway, it is the standard tool used by the plate reconstruction community.

Core Features

Interactive plate reconstruction to any geological time
Rotation file editing with plate hierarchy trees
Import of geological, geophysical, and paleomagnetic data
Animation of plate motions through time
Export of reconstructed geometries and velocity fields

Key Data Models

Müller et al. (2019): global rotation model back to 250 Ma
Seton et al. (2012): ocean floor age grid and spreading history
Merdith et al. (2021): full-plate model back to 1 Ga
Matthews et al. (2016): global plate velocities since 410 Ma

The Supercontinent Cycle

Plate reconstructions reveal that continental masses periodically assemble into supercontinents and then fragment. This supercontinent cycle operates on a timescale of 400–600 million years and is one of the most fundamental patterns in Earth history.

~1.0 Ga

Rodinia

The first well-constrained supercontinent. Laurentia (proto-North America) occupied a central position. Reconstructed primarily from paleomagnetic data and geological correlations of Grenville-age (~1.0 Ga) orogenic belts. Rodinia began to fragment around 750 Ma, coinciding with Snowball Earth glaciations.

~600 Ma

Pannotia (Greater Gondwana)

A short-lived supercontinent formed by the assembly of Gondwana from East and West Gondwana fragments. Some workers dispute whether Pannotia was a true supercontinent or merely a close clustering. It fragmented rapidly, with Laurentia rifting away by ~540 Ma to open the Iapetus Ocean.

~320 Ma

Pangaea

The most recent and best-constrained supercontinent. Formed by the collision of Gondwana and Laurussia (Laurentia + Baltica) during the Variscan/Alleghenian orogeny. Maximum assembly around 300–250 Ma. Pangaea began rifting in the Triassic (~200 Ma), splitting into Laurasia (north) and Gondwana (south) separated by the Tethys Ocean.

~250 Ma?

Amasia (Future Supercontinent)

Models predict the next supercontinent will form in ~200–250 Myr. The “Amasia” model (Mitchell et al., 2012) predicts assembly over the Arctic, closing the Arctic Ocean and the Caribbean. Alternative models include “Novopangaea” (closure of the Pacific) and “Aurica” (closure of both Atlantic and Pacific). The outcome depends on whether the Atlantic develops subduction zones.

Pangaea Breakup: A Reconstruction Summary

Stage Rotation for Atlantic Opening

\[ R_{\text{stage}}(83 \to 0 \text{ Ma}) = R_{\text{finite}}(0) \cdot R_{\text{finite}}(83)^{-1} \]

The stage rotation for the post-Cretaceous Normal Superchron opening of the South Atlantic is computed from the difference of finite rotations at anomaly 34 (83.6 Ma) and the present. The stage pole for this interval lies near 62°N, 36°W with a rotation angle of ~24°.

~200 Ma: Initial Rifting

Central Atlantic Magmatic Province (CAMP) erupts. Rifting separates North America from Africa. The central Atlantic begins to open. Tethys Ocean dominates equatorial region.

~160 Ma: Gondwana Fragmentation

East Gondwana (India, Australia, Antarctica) separates from West Gondwana (Africa, South America). Madagascar rifts from Africa. Indian Ocean begins to form.

~130 Ma: South Atlantic Opens

Africa and South America begin separating. Paraná-Etendeka flood basalts mark the onset of rifting. The South Atlantic opens from south to north like a zipper.

~80–50 Ma: India's Sprint

India races northward at up to 18 cm/yr, the fastest plate motion ever recorded. Deccan Traps erupt at ~66 Ma. India collides with Eurasia at ~50–55 Ma, initiating the Himalayan orogeny.

← 3.3 Triple Junctions Part 4: Divergent Boundaries →