Analysis of plate spin motion and its implications for strength of plate boundary

A unique feature of the Earth is active plate tectonics (Schubert et al. 2001), involving rigid plates that interact at “soft” boundaries, the nature of which allows the relative motion between plates. Significant differences between the physical properties of plates and plate boundaries (e.g., strength and rheological properties) are critical to plate dynamic motions and contribute to the “toroidal motions” of the plates (e.g., Bercovici 2003; Moresi and Solomatov 1998), especially the significant strike-slip and spin motions of the plates (e.g., Bercovici and Wessel 1994; Cadek and Ricard 1992; Hager and O’Connell 1978; Lithgow-Bertelloni and Richards 1993; O’Connell et al. 1991; Olson and Bercovici 1991). In the absence of such differences, plate motions would likely be dominated by “poloidal motions,” i.e., velocity fields without vertical vorticity at the Earth’s surface.

The high toroidal/poloidal kinetic energy ratio, which characterizes plate tectonics, has been used as a criterion to evaluate the accuracy of three-dimensional simulations of mantle convection that naturally reproduce the surface motions of plate tectonics (e.g., Richards et al. 2001; Tackley 2000a, b). However, the mechanism of the plates’ motions is not well understood at present (Bercovici 2003; Lithgow-Bertelloni and Richards 1993). If plate boundaries were sufficiently “hard” (i.e., rigid) to suppress relative plate motions, then the toroidal motion could not be generated. As the strength of plate boundary decreases, an appreciable toroidal motion begins to occur (Zhong et al. 1998), indicating that the mechanical strength of plate boundaries plays a key role in the plate dynamics of the Earth. Accordingly, a number of forward numerical simulations have examined various physical properties of the plates and their boundaries to reproduce Earth-like surface motions (e.g., Foley and Becker 2009; Lenardic et al. 2006; Richards et al. 2001; Tackley 2000a, b; van Heck and Tackley 2008).

Here, we adopt a different approach: Based on the observations of plate motion, we attempt to find the relationship between the motions and the mechanical strength of plate boundaries. We begin with a toroidal–poloidal decomposition of plate motion, based on the latest plate configurations (Argus et al. 2011; Bird 2003), to capture the global features of the velocity field. Then, we examine the motions of individual plates, particularly their spin motions, which may determine the constraints on the strength of plate boundaries, using a force balance approach.

The result of the spherical harmonic expansion is shown in Fig. 1. Although O’Connell et al. (1991) argued that the ratio of the amplitude of the toroidal to poloidal spectra is nearly constant for \(l<32\) (at \({\sim }0.8\)), our new result, applicable to higher harmonics, shows that the amplitude ratio is broadly trends toward higher values for \(l > 20\), which corresponds to a scale \({<}1000\) km, and is demonstrated by the approximate lines in Fig. 1.

To estimate the error of this result, caused by the uncertainties in plate motions, we calculated the toroidal/poloidal ratio of another plate model, NNR-MORVEL (Argus et al. 2011). The result is indicated by the red line in Fig. 1 and demonstrates that there is a little difference in the toroidal/poloidal ratio between the plate models, indicating that the error of plate motion in a plate model is negligible and the trend of toroidal/poloidal ratio in this analysis is robust.

The increase in the ratio of the amplitude of toroidal to poloidal spectra is probably the result of two factors: one is the systematic increase in toroidal motion, such as plate spin motion, with decreasing plate size; the other is related to plate geometry, such as systematic changes in the aspect ratio of plates with varying plate sizes (Olson and Bercovici 1991). To quantitatively examine the effects of plate geometry and motion on the toroidal/poloidal ratio, we focused on individual plates and investigated variations in the geometries and motions of plates with variations in plate size.

Although Eq. (2) is obtained for a highly idealized geometry, this relationship is valid for slightly more generalized cases as shown in Fig. 2, in which the relationship between the aspect ratio, plate size and the toroidal/poloidal spectra has been calculated and plotted. Figure 2 shows that the higher aspect ratio increases the toroidal/poloidal ratio, especially at high degrees of spherical harmonics (b, c and d in Fig. 2), and such increased toroidal/poloidal ratios occur at higher spherical harmonic degrees (i.e., finer scale of motion) for the smaller plate (c in Fig. 2). Therefore, \(\gamma \) in Eq. (2) can be regarded as a good measure to investigate the influence of plate geometry on the toroidal/poloidal ratio. Accordingly, we estimate the lengths of each plate in the directions parallel and normal to the observed plate motion. Figure 3 shows that although the aspect ratio of five plates exceeds 9 around the plate size of 1000 km, the aspect ratio does not systematically vary with plate size, indicating that aspect ratio is not the primary cause of the systematic increase in the toroidal/poloidal ratio observed in Fig. 1.

Subsequently, we examined individual plate motions with respect to plate size. For the analysis, we divided plate motions into two types, spin motion and straight motion. The plate spin motion generates a shear motion at the plate boundary and is associated with toroidal motion, whereas straight plate motion results in poloidal motion at both subduction zones and ridges, and toroidal motion along transform faults. Therefore, the ratio of spin motion to straight motion can be used as an index of the toroidal/poloidal ratio.

To obtain the spin and straight motions of individual plates, we divided the Euler vector of each individual plate into two components: a vector that passes vertically through the geometric center of the plate, which is related to the spin motion and has a magnitude (i.e., angular velocity) defined as \(\omega _C\), and a vector perpendicular to the first vector, which passes through the Earth’s center and is related to the motion along a great circle, and whose magnitude is defined as \(\omega _G\). For plates in the PB2002 dataset, plots of \(\omega _C\) and \(\omega _G\) as functions of plate size (Fig. 4) show that \(\omega _C\) generally decreases with increasing plate size, whereas \(\omega _G\) is roughly constant between 0.1 and 1\(^\circ \)/Myr. As a result, for plate sizes \({<}{\sim }1000\) km, the difference between \(\omega _C\) and \(\omega _G\) is large (Fig. 4). This difference can induce variations in the toroidal/poloidal ratio at high spherical harmonic degrees (Fig. 1). Based on these differential variations, we discuss the mechanisms and their corresponding force balance for the plate motions.

In this context, \(\omega _C\) of a slab-free plate, which is indicated by the blue circles in Fig. 5, generally increases with decreasing plate size. One notable feature in Fig. 5 is the rapid change in \(\omega _C\) around the critical plate size of \({\sim }1000\) km: above the critical size, \(\omega _C\) values are generally less than the average rotation rate of the global lithosphere (i.e., the net lithospheric rotation; Ricard et al. 1991), of \({\sim }\)0.43\(^\circ \)/Myr, as based on a hotspot reference frame; below the critical size, however, except for two plates, i.e., the Panama plate and the Shetland plate, the motions of which are not well determined in PB2002 model, all \(\omega _C\) values exceed 0.43 \(^\circ \)/Myr.

It should be noted that the R-\(\omega _C\) variation is not significantly affected by the choice of different plate models as shown in Fig. 5 [i.e., Pb2002 (Bird 2003), NNR-MORVLE (Argus et al. 2011) and GSRM v2.1 (Kreemer et al. 2014) that include both no-net-rotation and hotspot reference frames for the data acquired by several methods representing different timescales. See “Additional file 2: Table S1” for details]: i.e., almost all the large plates (plate size \({>}{\sim }1000\) km) show negligibly small \(\omega _C\) less than the net-rotation rate (0.43), and the rotation direction (clockwise or counterclockwise) may vary depending on the reference frame chosen (as indicated by large error bars with downward arrows in Fig. 5), whereas the large \(\omega _C\) is seen only for small plates irrespective of the chosen frame (Fig. 5).

In order to test such a possibility, we examine the relationship between R and \(\sigma _\mathrm{PB}\) (the plate boundary shear stress driving the spin motion) based on Eq. (5) and observed \(\omega _C\) as shown in Fig. 6, assuming \(\mu = 10^{20}\) Pa s, \(D_\mathrm{PB} = 40\) km, \(D_\mathrm{RS} = 60\) km, and \(\lambda = 0.25\). The uncertainties associated with these assumptions will be discussed later. It is worth noticing that \(\sigma _\mathrm{PB}\) for plates with low rotation rates (the large plates, in general) have extremely large uncertainties, indicated by the error bars with downward arrows in Fig. 6, and should be regarded as upper bounds.

The dashed line in Fig. 6 represents the expected driving stress induced by the motion of the Pacific plate (corresponding to the dashed line in Fig. 5), which limits the upper bounds of \(\sigma _\mathrm{PB}\) for the small plates. The small plates plot along the dashed line are located next to (or very close to) the fast-moving Pacific plate and are expected to have a high \(\sigma _\mathrm{PB}\) as in Fig. 6, whereas other small plates located next to plates with slower velocities are expected to have a lower \(\sigma _\mathrm{PB}\), which is also seen in Fig. 6. Within this context, in order to discuss the maximum stress that the plate boundary can sustain, the upper bound of \(\sigma _\mathrm{PB}\) and its variation with plate size are thought to be more important than the overall data distribution. For large plates, \(\sigma _\mathrm{PB}\) is appreciably lower than the prediction of the dashed line, indicating that rheological weakening or yielding operates on the boundaries of large plates, irrespective of the choice of reference frame (Fig. 6).

For small plates (the plate size \({<} {\sim } 1000\) km, i.e., \(R < 500\) km), the shear stress driving the spin motion increases with R (e.g., the dashed line in Fig. 6), and above a critical size \(R_\mathrm{c}\), the stress becomes too large to transmit the stress across the boundary, causing rheological weakening or yielding. This critical stress is regarded as the strength of the plate boundary. From Fig. 6, the critical stress is estimated to be \({\sim }10 - 20\) MPa for plate sizes between \({\sim }350{-}630\) km. Accordingly, the rotation rate of plates with \(R > R_\mathrm{c}\) is small as compared to the dashed line in Fig. 5.

It can be confirmed that the spin rates of small plates are higher than those of large plates, which is the overall result, from Fig. 7, which shows that many small plates along the “sides” (i.e., along strike-slip boundaries) of large fast-moving plates have high spin rates with a rotation direction (clockwise or counterclockwise) consistent with the nearly straight motions of large plates that subduct along their margins. This mechanism has been suggested for several individual microplates (e.g., Schouten et al. 1993). Figure 7, for example, demonstrates that in the southwestern Pacific, the Pacific plate (PA) excites spin motion of the Niuafo’ou plate (NI) and that the Australia plate (AU) induces spin motion of the Tonga plate (TO) and the Kermadec plate (KE). Along the East Pacific Rise, the Pacific plate and the Nazca plate (NZ) drive motions of the Easter plate (EA) and the Juan Fernandez plate (JZ). Although some of the abovelisted small plates, based on PB2002 plate model, are located within deforming zones identified by Kreemer et al. (2014) and could be inappropriate to consider them as rigid plates to define the spin rate (e.g., KE), the overall configuration remains unchanged, including large spin motions of NI, EA, and JZ as shown in Fig. 7.

These features, represented in Fig. 7, suggest that although some regions (especially Southeast Asia) exhibit complex spin directions probably due to interactions among the small plates, large fast-moving plates with subducting slabs induce spin motion in adjacent small plates through interactions along plate boundaries, which supports the idea presented above that \(T_{PB}\) drives the plate spin motions.

In previous studies, the strength of a plate boundary was estimated based on seismic observations, particularly the spatial mapping of earthquake focal mechanisms and their corresponding temporal changes before and after a large earthquake; e.g., Hasegawa et al. (2012) used high-resolution data on the approximately 4000 earthquake focal mechanisms in northeast Japan between 2003 and 2011 and found that the 2011 Tohoku earthquake almost completely released the accumulated stress along the plate boundary, and they estimated the stress at release to be as small as 5–15 MPa, suggesting that the presence of water weakened the plate boundary fault. Hardebeck and Hauksson (2001) used the focal mechanism data of approximately 50,000 earthquakes along the San Andreas Fault, mainly between 1981 and 1999, including the 1992 Landers earthquake, and estimated the strength of the fault to be about \(20 \pm 10\) MPa, due in part to the low mechanical strength of smectite (Carpenter et al. 2011). In this study, although the estimated strength of 3–75 MPa may be regarded as a global average strength for a number of plate boundaries of various types, including convergent and transform boundaries, the estimated strength is consistent with a stress level deduced from high-resolution seismic observations of specific areas, as mentioned above.

To naturally reproduce plate-like structures and motions as part of a mantle convection process, including the case of rigid plates with soft plate boundaries, three-dimensional numerical simulations have been used to investigate critical conditions and requirements, especially those concerning rock rheology. Tackley (2000b) and Richards et al. (2001) estimated the required yield stress of a plate to reproduce Earth-like plate motion on the basis of surface velocity fields and obtained results of 17–170 and 50–150 MPa, respectively. In addition, Bercovici (1993, 1995, 2003) suggested that a pseudo-stick-slip rheology, in which the stress decreases with increasing strain rate after yielding, can reproduce plate-like motions, in particular a high toroidal/poloidal kinetic energy ratio of up to 0.8, which is consistent with ratios observed in previous studies (Bercovici and Wessel 1994; Hager and O’Connell 1978; O’Connell et al. 1991) as well as in this study (Fig. 1). In this context, after exceeding the yield stress of plates, the strain can be concentrated to form a plate boundary composed of a “damaged” weak zone (e.g., Bercovici and Ricard 2014), where the stress level is significantly reduced, possibly to the range estimated from seismology and this study, especially when water is present to weaken the plate boundary.

The exact rheology and the physical–chemical state of plate boundary is a vital problem that will help understand the mechanisms of plate tectonics (e.g., Bercovici and Ricard 2014; Gordon 1998, 2000). At present, it is difficult to constrain the exact rheology from the approach in this study; however, by combining with other approaches, such as seismic and geodetic observations on both rigid plates and deformation zones (Gordon 1998, 2000; Kreemer et al. 2014), field and laboratory studies on rock and fault rheology (e.g., Kohlstedt et al. 1995; Sibson 2003), and numerical simulation of combined plate motion and mantle convection (e.g., Bercovici 2003; Richards et al. 2001; Tackley 2000b), tighter constraints can be obtained to quantify the Earth’s dynamics with regard to plate tectonics.

The main objective of this study was to constrain the driving forces of plate motion, especially the plate spin motion, for which we analyzed the relationship between toroidal–poloidal components, plate geometry, plate size, spin motion, and their geographical distribution. The following results were obtained. First, a continuous increase in the toroidal/poloidal ratio at high spherical harmonic degrees (\({>}20\) up to 1000; Fig. 1) arises from an increase in the spin rate of individual plates (Fig. 4), rather than a consequence of geometrical factors (e.g., a systematic change in aspect ratio with plate size; Fig. 3). Second, spin motion of plates without slabs decreases at plate sizes of \({\sim }1000\) km and greater (Fig. 5), which indicates the strength for plate boundaries (Fig. 6). Third, the geographical distribution of spin motion (Fig. 7) suggests that large plates with subducting slabs drive the spin motion of surrounding smaller plates, similar to gears that transmit the shear stress induced by straight motion of large plates. Finally, from the dynamic equilibrium of spin motion at the critical plate size, we obtain the relationship between the strength of the plate boundary and the viscosity of the asthenosphere. Assuming the viscosity and thickness of the asthenosphere to be \(10^{19}\)–\(10^{20}\) Pa s and 60–140 km, respectively, we roughly estimated the strength of the plate boundary to be 3–75 MPa, which is comparable to the stress level estimated from several seismological observations, including those in NE Japan associated with the 2011 Tohoku earthquake.