Plate Reconstruction Interpolation - Appendix I

Appendix I: Derivation of instantaneous pole and angle of plate motion from analytic finite rotation functions  (Plate Reconstruction Interpolation)

(Based on Smith, E. G. C., 1981, Calculation of poles of instantaneous rotation from poles of finite rotation, Geophysical Journal of the Royal Astronomical Society, 65, 223-227, and Pilger, R. H., Jr., 2003, Geokinematics: Prelude to Geodynamics, Springer-Verlag, Berlin.)

The relation to be derived is:

w_T = l_T’ W_T + sin l_T W_T’ + (cos l_T –1 ) W_T X W_T’

in which:

     w_T = instantaneous angular velocity pseudovector for time T; magnitude is equal to the instantaneous rotation rate in radians,

l_T = total finite rotation angle for time T,

W_T = unit vector representation of the finite pole of rotation for time T,

X denotes the vector cross product, and the prime (‘) denotes the first time derivative.
Derivation follows the jump.

Plate Reconstruction Interpolation - Table 1

Table 1 (Plate Reconstruction Interpolation):

Plate Reconstruction Interpolation

A New Approach to Plate Reconstruction Interpolation and Plate Kinematic Inference

1/22/2003

Introduction

The reconstruction and kinematic models of plate tectonics conveniently rely on a theorem of Leonid Euler: A kinematically rigid displacement confined to the surface of a sphere (equivalently: a displacement with one fixed point – the center of the sphere) can be parameterized as a rotation around an axis passing through the center of the sphere (and origin of a Cartesian and/or spherical coordinate system – the one fixed point of the system; e.g., Weisstein, 1998). More conveniently, the axis of rotation corresponds with the latitude and longitude of the two points at which it intersects the sphere’s (the earth’s) surface -- the poles, instead of the less-easily visualized direction cosines. 

Structural Geology of the Shallow Earth

Structural Geology of the Shallow Earth (<1000 km deep)
December 31, 2003
Structural geology has as its principal task the inference of the geometry and evolution of rock bodies (sedimentary, metamorphic, and volcanic strata and magmatic intrusives) contemporaneous with or subsequent to their deposition or emplacement. Traditionally, structural geology has been restricted to study of rock outcrops at the Earth’s surface. Over the past few decades, surface observations have been enhanced by incrementally higher resolution geophysical observations, especially reflection seismology at shallow depths and refraction and reflection tomography at depth.

Three Hotspot Reference Frames

Three Hotspot Reference Frames – Shallow and Lithospherically Controlled – Reflect “Mesoplates,” a Plate Tectonics Heuristic
May 1, 2003 (Minor edits, December, 2010)
The hotspot reference frames of the Atlantic-Indian Ocean and the Pacific Ocean are kinematically distinct. Relative motion of the two reference frames in the Late Cretaceous and Early Tertiary corresponds with motion of the American plates in the Atlantic-Indian Ocean frame, suggesting a lithospheric control on the reference frames. A third reference frame, beneath Iceland and Eurasia, cannot be satisfactorily quantified at this time, but the inability to fit Iceland in either of the two other reference frames implies its existence.
The two hotspot reference frames can be shown to be relatively shallow. Hotspot traces, cross-grain oceanic gravity lineations of the Pacific Ocean reflect a shallow hotspot reference frame. Intracontinental stress fields of North America correspond with the Atlantic-Indian Ocean reference frame, also reflecting a shallow reference frame.

Geokinematics and Lithoplate Structure: Controls on Hotspot Origin and Evolution

PDF Link
Geokinematics and Lithoplate Structure: Controls on Hotspot Origin and Evolution
December 2, 2003
Rex H. Pilger Jr.
Abstract
Whatever the origin of “hotspots”, the reference frames they define are shallow, based on several distinct lines of evidence: (1) Occurrence of minor hotspot chains in the Pacific and South Atlantic appears to be controlled by plate age (therefore, thickness) and sublithospheric heterogeneities. (2) Similarly cross-grain gravity lineations in the Pacific seem to be age-controlled. (3) Intracontinental stress fields are consistent with hotspot kinematic models.

Geokinematics and Lithoplate Structure - Figures

Geokinematics and Lithoplate Structure - Figures 

Figure 1. a. Hypothetical hotspot trace, map view; trace includes an apparent distinct change in plate motion relative to hotspot reference frame as well as slight perturbations in location.

Lithoplates and Mesoplates: Expanding the Scope of Plate Tectonics

Original: June 3, 2005
Abstract
The existence of two distinct hotspot reference frames in motion relative to one another now appears to be conclusively demonstrated over the past 80 m.y. Lithospheric plate (“lithoplate”) motions relative to the two reference frames are recorded not only by hotspot traces, but by intracontinental stress field orientations, oceanic gravity lineations, and reconstructions of Caribbean island arcs. The reference frames represent plate-like layers within the upper mesosphere, termed “mesoplates”. Mesoplates, of which three appear to exist, are in motion relative to one another as well as relative to overlying lithoplates. Subduction zones appear to define part of the boundary between mesoplates and depending on the motion of the upper lithoplate of a zone, control their relative motion. Mesoplates are not merely the habitat of embedded hotspots, they may also be the environment in which plumes responsible for the hotspots develop, in response to passive or active lithospheric thinning and consequent isostatically-driven depressurization melting.

Lithoplates and Mesoplates: Figures

Figures for Lithoplates and Mesoplates: Expanding the Scope of Plate Tectonics

Figure 1. Cartoon illustrating principal features of mesoplate-lithoplate hypothesis, modified after Pilger (2004). Two mesoplates, four large lithoplates, one microlithoplate, and one hotspot (with trace) are shown. Bullseye symbol implies relative fixity of the indicated mesoplate and microlithoplate; that is, other objects are moving relative to them. Note divergence of Mesoplate B relative to A in the vicinity of the subduction zone bounding Lithoplates 2 and 3, with low-angle subducting Lithoplate 3 filling the gap between the mesoplates. Note convergence of Mesoplates A and B in the vicinity of the Lithoplate 4 subduction zone beneath the microlithoplate and Lithoplate 1, forming a “double” subduction zone. Spreading center between Lithoplates 3 and 4 occurs entirely above Mesoplate B; isostatic rise of asthenosphere results in partial melting of Mesoplate B beneath ridge, progressively converting mesoplate to asthenosphere. Hotspot a is moving relative to Mesoplate A, while Lithoplate 3 is moving relative to both Mesoplates B (with embedded hotspot) and A.

Caribbean-EAFRC Graphics

Figures for Caribbean Back-Arcs and East African Rift Volcanism: Mesoplate Tracers

Figure 1. Mesoplate boundaries (dotted), modified from Pilger (2003a, b).

Caribbean Back-Arcs and East African Rift Volcanism: Mesoplate Tracers

PDF Link
Abstract
Several lines of evidence imply that hotspots are embedded within shallow, kinematically rigid layers in the outer mesosphere. The existence of such “mesoplates” in motion relative to one another is enhanced with recognition of stationarity of the Eastern Caribbean island arc relative to the Atlantic-Indian hotspot frame (Tristan mesoplate) over the past ~70 m.y. In addition, inception of volcanism along the East African Rift over the past ~60 m.y. occurred at the same north-to-south rate as African motion in the Tristan frame. Embedding of hotspots in mesoplates is consistent with a top-down origin of the “plumes” inferred to produce hotspots, especially in light of recent tomographic evidence for variable depth of origin of plumes.

Fractal Plate Reconstruction Table 2

Age (Ma)		2.581	6.033	11.040	19.722	26.154	28.715	31.116	33.738	37.771
Latitude		12.395	11.906	11.049	12.850	13.126	13.039	13.032	13.303	14.622
Longitude		41.738	40.366	37.655	32.877	34.052	34.616	34.722	34.352	32.585
Angle		1.001	4.057	6.781	12.168	15.947	17.272	18.670	20.360	22.499
SAF	I	0.4982	0.5018	0.5242	0.5100	0.4970	0.4840	0.4851	0.4983	0.4887
	II	0.4664	0.4408	0.5585	0.5320	0.5254	0.5416	0.5398	0.5177	0.5025
	III	0.4652	0.5000	0.5601	0.5414	0.5302	0.5109	0.5007	0.4965	0.4749
	IV	0.5002	0.5078	0.5083	0.5279	0.5468	0.5502	0.5437	0.5303	0.5000
	V	0.5000	0.5242	0.5000	0.5038	0.5000	0.5000	0.5000	0.5000	0.5000
	VI	0.4890	0.4751	0.4907	0.5039	0.5336	0.5030	0.4863	0.4904	0.5000
	VII	0.5000	0.4730	0.5000	0.5000	0.5000	0.5000	0.5519	0.5000	0.4654
	VIII	0.5000	0.5000	0.5000	0.5000	0.5000	0.5515	0.5343	0.5186	0.5000
	IX	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5109	0.5576	0.5285
	X	0.5311	0.5343	0.4991	0.5248	0.5449	0.5526	0.5495	0.5370	0.5000
	FZs	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000	0.5000

Table 2. Final segment rotation parameters and asymmetry factors for magnetic isochron crossing ages, Australian and Antarctic oceanic plates. Partial rotation angle for Antarctic plate to the ridge is the asymmetry factor multiplied by the total rotation angle. For the Australian plate the partial rotation angle to the ridge is (1.0 – the asymmetry factor) multiplied by the total rotation angle. In the absence of data points, asymmetry factor is assumed to be 0.500. 

© 2010 Rex H. Pilger, Jr.
Return to article.