Broken Sheets—On the Numbers and Areas of Tectonic Plates
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Broken Sheets—On the Numbers and Areas of Tectonic Plates Bruce H. Wilkinson, Department of Earth Sciences, Syracuse University, Syracuse, New York, 13244, USA, [email protected]; Brandon J. McElroy, Department of Geology & Geophysics, University of Wyoming, Laramie, Wyoming, USA; and Carl N. Dummond, Department of Physics, Purdue University Fort Wayne, Fort Wayne, Indiana 46805-1499, USA ABSTRACT unpredictable, thus reflecting the inher- comprise a single continuous power law The sizes and numbers of tectonic ently complicated nature of processes distribution of size frequency, but with the plates are thought to record the impor- associated with their formation. proviso that a finite Earth surface area tance of plate division, amalgamation, and imposes an upper limit on larger plate INTRODUCTION destruction at divergent and convergent areas. Like Bird (2003), Morra et al. (2013) margins. Changes in slope apparent on log The outer brittle layer of the Earth con- concluded that plate sizes consist of large area versus log frequency plots have been sists of lithospheric plates that move over the and small populations, and that this differ- interpreted as evidence for discrete popu- relatively weak asthenosphere. Larger-scale ence in plate areas persists back in time at lations of plate sizes, but the sizes of litho- aspects of plate movement are the surficial least several tens of millions of years. spheric plates are also closely approxi- manifestations of mantle convection in the Mallard et al. (2016) employed spherical mated by a continuous density function in deeper Earth, but local processes of defor- models of mantle convection to examine which diameters of individual plates are mation may be only indirectly related to the geodynamical processes that drive the exponentially distributed; such size fre- regional stress fields. Because size frequen- tessellation of the Earth’s lithosphere and quencies are dependent only on the total cies from brittle fragmentation might be concurred with Bird (2003) and Morra et area and number of designated elements. manifest as self-similar (fractal) power laws al. (2013) that plate areas comprise several This implies that the spatial locations of (e.g., Davydova and Uvarov, 2013), it is distinct populations. Harrison (2016) pro- plate boundaries are controlled by a myr- important to know the relationships between posed an additional 107 plates as subdivi- iad of complicated and interrelated pro- lithospheric rheology and degree of frag- sions of the 52 plates proposed by Bird cesses such that the geographic occur- mentation, as well as the degree to which the (2003); he also interpreted several changes rence of any particular boundary is largely breakup of the lithosphere reflects the of slope in log-log plots of plate number indeterminate and thus spatially indepen- dynamics of mantle convection versus local versus area as reflecting the presence of dent of the proximity of other plate bound- interactions along plate boundaries. This is several size populations. aries. Observed breaks in slope on linear- particularly so as models of the evolution of While the designation of any particular ized size versus frequency plots are tectonic plates are extended over ever- region as a discrete “plate” is somewhat of merely coincidental and of themselves do increasing spans of geologic time (e.g., an evolving enterprise (e.g., Zhang et al., not support an interpretation of discrete Domeier and Torsvik, 2014; Matthews et al., 2017), the actuality of either single tectonic processes operating over distinct 2016; Merdith et al., 2017). (Sornette and Pisarenko, 2003) or multiple length scales. Although a purely random One approach to this question derives (Anderson, 2002; Bird, 2003; Mallard et distribution of plate boundaries also impli- from considerations of the numbers and al., 2016; Harrison, 2016) populations of cates a similar chance distribution of plate sizes of the tectonic plates. Anderson plate sizes is important to interpreting the sizes, some smaller plates are indeed clus- (2002), for example, noted that fracture manifestation of causative tectonic pro- tered along convergent boundaries in the patterns tend to self-organize such that cesses. If sizes of tectonic plates are read- southwestern Pacific. Such association of mud cracks, frozen ground, basalt col- ily characterized by a single frequency plates of similar (small) sizes suggests umns, and other natural features exhibit distribution, be it fractal or otherwise, an that locations of plate boundaries are best similar patterns. He argued that tectonic interpretation of multiple processes operat- described as reflecting nonhomogeneous plates therefore might consist of semi-rigid ing at distinct length scales is not sup- Poisson processes wherein probabilities larger polygons separated by (at times dif- ported. Conversely, if plate area-frequen- of reaching some plate boundary vary fuse) boundary zones of deformation sur- cies are best described by multimodal along any Earth-surface transect. Size faced by smaller elements. Bird (2003) distributions, then interpretations linking frequencies of continents, calderas, and presented a global data set interpreted as populations of large plates to processes many other geologic entities where embodying several fractal subpopulations occurring at convective length scales (e.g., dimensions are expressed as areal extent of plate sizes, each manifest as an approxi- Lenardic et al., 2006) and populations of exhibit similar size-frequency distribu- mately linear trend in log area versus log smaller plates to the generation of litho- tions, suggesting that lateral occurrences occurrence–frequency space. Sornette and spheric fragments at the edge of plate of their boundaries are also largely Pisarenko (2003) argued that plate sizes boundaries are entirely reasonable. GSA Today, v. 28, doi: 10.1130/GSATG358A.1. Copyright 2017, The Geological Society of America. CC-BY-NC. NUMBERS AND AREAS OF Plate Area (km2) TECTONIC PLATES 10,000 1,000,000 100,000,000 100 Detailed data on major tectonic bound- aries and areas of enclosed plates are sum- A n = 52 p = 0.045 % /km marized by Bird (2003), who presented the R2 = 0.87 characteristics and locations of 6,048 Bird (2003) utuna Manus Gurnis et al. (2012) F points that serve to define 229 boundary Shetland Galapagos segments delineating 52 lithospheric Fernandez plates. The relationship between plate area Juan 10 and frequency of occurrence is most con- B n = 20 South American Australian veniently represented as a cumulative fre- p = 0.025 % /km ceedence 2 Eurasian R = 0.93 Ex quency distribution in which plate area is North American plotted relative to size exceedance—the 10 Antarctic e South American number of plates equal to or larger than North American African any size in question; the Y-intercept East Antarctic Australian ceedenc Pacic defines the total number (e.g., 52) of data Eurasian 1 Ex values (Fig. 1). E = n e(-Ap2)0.5 African Breaking and annealing P = 0.5 Nπ/T0.5 THE BROKEN SHEET FUNCTION Resamplings of Broken Sheets Pacic 1 Average breaking & annealing The general curvilinear form of log-log 1,000,000 100,000,000 Broken Sheet Function plate area frequency distributions is simi- 2 Inferred log-linear trends lar to size-frequency distributions for some Plate Area (km ) other area mosaics that include regions of Figure 1. Areas of tectonic plates relative to exceedance, the number of plate areas equal to or like sediment (lithotopes) across deposi- greater than the x-axis values. (A) Plate areas from Bird (2003). (B) Areas from Gurnis et al. (2012). Division into three apparent subpopulations in (A) (n = 52) and two subpopulations in (B) (n = 20) is tional surfaces (Wilkinson and based on seemingly linear trends (straight red lines). Light blue lines in (A) are 500 sets of plate Drummond, 2004), regions on global geo- areas (n = 52) resampled from a stable model time series wherein two random plates are annealed and another two randomly divided over thousands of iterations (see text). Heavy blue line in (A) is the logic maps (Wilkinson et al., 2009), sizes average area of many realizations of such annealed and broken plates. Light brown lines in (B) are of geopolitical subdivisions (McElroy et 500 sets of plate areas (n = 20) resampled from the broken sheet density function. Heavy brown line al., 2005), and taxonomic divisions of in (B) is the ideal broken sheet size frequency distributions predicated by the number of designated plates (20) and the total area of the sphere on which they exist (4 π steradians; ~510 × 106 km2). White organismal morphospace (Wilkinson, lines in (A) and (B) are the two series whose areas, by chance, are closest to observed sizes. Since 2011). Size-frequency distributions for there is no inherent division of sizes in these models, any subdivision based on the perception of differing slopes for straight-line segments is spurious. these tessellations reflect the partitioning of the total area into mosaics of sub-ele- ments wherein locations of boundaries, and therefore the sizes of these elements, are statistically independent. exponentially distributed. The broken per unit length of transect, itself expressed Conceptually, if the sizes of sub-ele- sheet distribution (e.g., Fig. 1) is the form as Equation (2), ments are represented by linear distances that results when sizes of randomly parti- p = 0.5 np/A0.5, (2) across them rather than areas, the distribu- tioned sub-elements are plotted as areas, rather than distances. The surface of the tion of distances between boundaries is the and A is the total surface or “sheet” area Earth can therefore be described as being same as that arising from the classic one- being divided (for Earth, ~510 × 106 km2). subdivided into tectonic plates such that dimensional “broken stick” model of ran- Comparisons to the 52 plate areas from dom linear subdivision proposed by plate sizes, as measured by the linear dis- Bird (2003) and the 20 plate areas from MacArthur (1957), who suggested that tances across them, are exponentially dis- Gurnis et al.