## Summary

Diffuse plate boundaries, which are zones of deformation hundreds to thousands of kilometres wide, occur in both continental and oceanic lithosphere. Here, we build on our prior work in which we described analytic approximations to simple dynamical models that assume that the vertically averaged viscous force resisting deformation in diffuse oceanic plate boundaries (DOPBs) is described by either a linear Newtonian viscous rheology or a yield-stress (high-exponent power-law) rheology.

An important observation is that the poles of relative rotation of adjacent component plates tend to lie in the diffuse plate boundary that separates them. A key cause of this tendency is that a faster spin is needed to balance a component of torque through the middle of a diffuse plate boundary than to balance an equal component of torque lying 90° from the middle of the diffuse boundary. The strength of that tendency depends on rheology, however, with the tendency being stronger for a yield-stress rheology than for a Newtonian viscous rheology. For the special case of the pole of rotation lying outside of and along the strike of the boundary, these large differences can be simply explained in terms of the distribution of boundary-perpendicular normal forces acting across the boundary. In the Newtonian case, the distribution of forces has an along-strike gradient that can balance a component of torque about the middle of the boundary, while in the yield-stress case, the distribution of forces has zero along-strike gradient and cannot balance a component of torque about the middle of the diffuse plate boundary.

To expand our analysis to intermediate power laws of geophysical interest (i.e. power-law exponents of 3 to 30), as well as to investigate more thoroughly the behaviour for a high-exponent power law, we numerically integrate the force distribution to obtain the torques. Results for intermediate power laws resemble the yield-stress rheology much more than they resemble the Newtonian rheology and depend only weakly on the width of the deforming zone. To quantify the probability that a pole of rotation lies in a diffuse plate boundary, we numerically integrate the expectation assuming that all orientations of torque are equally probable. For a power-law exponent of n= 10: 49 per cent of possible torque orientations produce angular velocities outside the diffuse plate boundary if the boundary is 55° long (similar in length to the boundary between the Nubian and Somalian component plates); 21 per cent of possible torque orientations produce angular velocities outside the diffuse plate boundary if the boundary is 30° long (similar in length to the boundary between the Indian and Capricorn component plates and to that between the Capricorn and Australian component plates); and 6 per cent of possible torque orientations produce angular velocities outside the diffuse plate boundary if the boundary is 15° long (similar in length to the boundary between the North American and South American component plates and to that between the Macquarie and Australian component plates). These results reinforce the prior conclusion that the pole is more strongly locked into the boundary if a DOPB is short than if it is long. For all boundary lengths, but even more so for short boundaries, the relationship between angular velocity and torque depends only weakly on the power-law exponent of the rheology as long as n≥ 3. From this, we conclude that orientation of the relative torque across a DOPB can be inferred from the location of the pole of rotation without precise knowledge of the appropriate power-law exponent.

## Introduction

It is widely assumed that oceanic tectonics is well described by a model of rigid plates separated by narrow plate boundaries such as transform faults, ridges and trenches, across which all deformation occurs. Although much deformation is localized in these narrow boundaries, broad zones of deformation (diffuse plate boundaries) nevertheless occur within traditionally defined plates across which displacements take place at rates that are generally slower than those across traditional narrow plate boundaries (2–16 mm yr−1 across diffuse plate boundaries as opposed to 12–160 mm yr−1 across mid-ocean ridges and transform faults, and 20–100 mm yr−1 across trenches; Gordon 1995, 1998).

These diffuse oceanic plate boundaries (DOPBs), with linear dimensions of 1000 km or more, include the boundary between the North American Plate and South American Plate (Ball & Harrison 1970; Argus 1990; Gordon 1998), which are component plates that are part of the American composite plate (Gordon 1998), the boundaries that separate the Indo-Australian composite plate into the Indian, Capricorn, Australian and Macquarie component plates (Wiens et. al 1985; DeMets et. al 1988; Gordon et. al 1990; van Orman et. al 1995; Royer & Gordon 1997; Conder & Forsyth 2001; Cande & Stock 2004), and possibly the oceanic portion of the boundary that separates the African composite plate into the Nubian and Somalian component plates (DeMets et. al 1990; Jestin et. al 1994; Chu & Gordon 1999; Lemaux et. al 2002; Horner-Johnson et. al 2005). A key kinematic observation of these boundaries is that the pole of relative rotation of the plates flanking the diffuse plate boundary lies in the boundary itself (Royer & Gordon 1997; Gordon 1998; Chu & Gordon 1999).

Using analytical models, we previously addressed the question of why the pole tends to lie in the boundary (Zatman et. al 2001). We approximated the vertically averaged rheology of deforming oceanic lithosphere as a power-law fluid (DeBremaecker 1977), for which . In this expression, is strain rate, τ is deviatoric stress and n is the power-law exponent. With this approximation, analytical models of diffuse boundary dynamics can be constructed for the Newtonian and plastic (i.e. yield-stress) end-member rheologies (i.e. n= 1 and n→∞; Zatman 2001). These models show that the pole of relative rotation between component plates is unlikely to lie outside their mutual diffuse boundary irrespective of rheology. Moreover, the pole is less likely to lie outside their mutual diffuse plate boundary, especially in the along-strike direction, for the case in which n→∞ than for the case in which n= 1.

Neither end-member is realistic, however. A more quantitative analysis of the deformation across diffuse oceanic boundaries and of the torque balances on adjacent component plates requires us to examine numerical models for intermediate rheologies. Here, we examine these questions more thoroughly. In particular, we examine how a more realistic vertically averaged rheology affects the probability that the pole of rotation lies in the diffuse plate boundary. Our results show that the relation between the torque and angular velocity is generally insensitive to the vertically averaged rheology of oceanic lithosphere and that realistic rheologies are much more like the yield-stress case than like the Newtonian case. This result suggests that the orientation of the torque that one component plate exerts on the other (and vice versa) can be estimated from their relative angular velocity without precise knowledge of the appropriate power-law exponent.

In the following sections, we first examine the geometrical effect that tends to restrict the location of the pole of rotation and present a qualitative explanation of why the pole of rotation is more likely to lie in the diffuse plate boundary for a high power-law rheology than for a Newtonian rheology. We then turn to the numerical models and results.

## Geometrical Effects

A key reason why the pole of rotation tends to lie in the diffuse plate boundary is geometrical (Zatman et. al 2001; Fig. 1). Assumed-rigid component plates A and B exert a torque on one another that is transmitted through the diffuse plate boundary. If the torque is large enough to cause failure of the entire upper lithosphere, which fails brittlely or semi-brittlely (Kohlstedt et. al 1995), then the lithosphere undergoes distributed permanent deformation thereby creating a diffuse plate boundary. Gordon (2000) has argued that the vertically averaged rheology of deforming oceanic lithosphere of the diffuse plate boundary is well approximated as a power-law fluid, as is the case for deforming continental lithosphere (Sonder & England 1986).

Figure 1

Cartoon showing an idealized composite plate comprising two components plates (A and B) and a diffuse oceanic plate boundary (labelled D.O.P.B.) Also shown are three axes of a right-handed Cartesian coordinate system. The angular velocity, ω, of component plate B relative to component plate A can be decomposed into three components, as can the torque, T, that A and B exert on one another through the DOPB.

Figure 1

Cartoon showing an idealized composite plate comprising two components plates (A and B) and a diffuse oceanic plate boundary (labelled D.O.P.B.) Also shown are three axes of a right-handed Cartesian coordinate system. The angular velocity, ω, of component plate B relative to component plate A can be decomposed into three components, as can the torque, T, that A and B exert on one another through the DOPB.

That the idealized diffuse plate boundary of Fig. 1 is located where composite plate AB is narrowest is no coincidence. The narrowing of the plate causes a concentration of stress, which leads to failure of the upper lithosphere. While the lithosphere in the neck is fluidized (in its vertically averaged rheology), lithosphere outside the neck behaves elastically (i.e. rigidly over geologic time) because the force per unit length is below the yield strength of the upper lithosphere. Thus, the idealized diffuse plate boundary of Fig. 1 is like the real-world diffuse plate boundary between the North American and South American component plates, which is located in a narrow neck between the Mid-Atlantic ridge and the Lesser Antilles trench in the American composite plate. Stress concentrations can occur by other mechanisms as well (e.g. Cloetingh & Wortel 1986; Coblentz et. al 1998) and strongly influence where diffuse oceanic plate boundaries occur in the Indian ocean (Gordon 2000).

It is convenient to consider the components of the relative torque and of the relative angular velocity in the coordinate system illustrated in Fig. 1. The z component lies in the middle of the diffuse plate boundary. The x component lies 90° from the z component along a great circle following the strike line of the diffuse plate boundary. The y component lies 90° from each of the first two components such that xy and z form a right-handed coordinate system (Fig. 1). Each component of torque external to the plate boundary is balanced by a torque across the plate boundary as a result of deformation in the boundary, which is in turn a result of rotation between component plate A and component plate B. Tz induces a component of angular velocity ωz. Similarly, Tx induces ωx and Ty induces ωy. ωi only depends on Tj if i=j.

Because the x and y components of ω and T are located at or near 90° from the deforming region of the diffuse plate boundary, a small spin about the x or y axes produces a much larger displacement across the diffuse boundary than the same spin about the z-axis (which lies in the diffuse boundary). For a diffuse plate boundary that varies only modestly in across-strike width, the strain rates in the boundary are approximately proportional to the local displacement rate between the two component plates. Hence, in the Newtonian case, the balancing torque (which is proportional to the strain rate) is expected to be proportional to the displacement rate. Thus, to balance a given component of torque about the x or y axes, only a relatively slow rate of spin about those axes is required.

For power-law fluids, as will be shown below, the pole of rotation is even more strongly confined to the diffuse plate boundary. For a power-law fluid, the balancing torque is proportional to the nth root of the strain rate and thus is expected to be proportional to the nth root of the displacement rate. Thus, the ratio of ωz to ωx or of ωz to ωy must be even larger for a power-law fluid, than for a Newtonian fluid, to balance comparable-sized components of T.

## Newtonian Versus Yield-Stress Rheology

For the rest of our analysis, we use the coordinate system elaborated in Fig. 2. The origin coincides with the centre of the Earth. A reference axis runs through the middle of the diffuse boundary, which is assumed to lie astride a great circle, GC. Although it is possible to generalize the geometry for kinked or irregularly shaped boundaries, we do not do so here. θ is the angular displacement about the centre of the Earth along GC and ψ is the angular displacement along a great circle normal to GC. If one thinks of GC as the equator of the spherical coordinate system we use, then θ is longitude and ψ is latitude. ω is the angular velocity of component plate B relative to component plate A. T is the associated boundary torque about the centre of the Earth acting on component plate A.

Figure 2

Model setup and geometry. A diffuse oceanic plate boundary (DOPB; shaded area) separates component plate A from component plate B. ψ is latitude and θ is longitude. The origin of the coordinate system (θ, ψ) = (0, 0) is taken to be the centre of the boundary and a great circle (GC) through the boundary is chosen along which ψ= 0, i.e. the equator of the coordinate system. ω is the angular velocity of component plate B relative to component plate A and T the associated torque that component plate B applies to plate A via the boundary.

Figure 2

Model setup and geometry. A diffuse oceanic plate boundary (DOPB; shaded area) separates component plate A from component plate B. ψ is latitude and θ is longitude. The origin of the coordinate system (θ, ψ) = (0, 0) is taken to be the centre of the boundary and a great circle (GC) through the boundary is chosen along which ψ= 0, i.e. the equator of the coordinate system. ω is the angular velocity of component plate B relative to component plate A and T the associated torque that component plate B applies to plate A via the boundary.

If Tz is comparable to both Tx and Ty, the geometrical effect discussed above largely explains the tendency for the pole of rotation between component plates separated by a diffuse plate boundary to lie in the boundary. For some extreme cases for which Tz is small compared with Tx or with Ty or with both, however, the pole location apparently depends strongly on rheology, giving qualitatively different results for n= 1, on the one hand, and for n→∞, on the other. Consider the case for which (Ty/Tx) ≪ 1 and the length of the diffuse plate boundary is 30°. When θT is 85° to 95° (i.e. ), θω is 15° or less for the plastic-like rheology (n→∞), which indicates that the pole of rotation is confined to the diffuse plate boundary. For the Newtonian case, however, θω can approach 90° (Zatman et. al 2001).

Fig. 3 illustrates, in terms of simple physics, why the pole of rotation is better confined in the boundary for a plastic rheology (n→∞) than for a Newtonian rheology (n= 1). The entire along-strike length of the boundary is shown. Two different poles of rotation, one in the DOPB and one outside of it, are considered. Both poles of rotation lie along the central strike line; thus, in both cases, all displacement is perpendicular to the central strike line. The left-most column of images shows the distribution of forces across the boundary for each of both poles and both rheologies, with the arrows representing the local contribution from a segment of the boundary to the force across the boundary.

Figure 3

Sketch of distribution of forces in an idealized diffuse oceanic plate boundary (DOPB) when the pole of rotation lies along the central strike line. Two pole locations are are considered: (i) when the pole lies in the boundary and (ii) when it lies outside the boundary. Two end-member rheologies are also considered: (i) Newtonian viscous (corresponding to the n= 1 end-member of a power-law fluid) and (ii) plastic (yield strength) behaviour (corresponding to the n→∞ end-member of a power-law fluid). Arrows show the distribution of force as one moves along strike in the DOPB (left-hand column of four figures) for a flat Earth model. The distribution of forces can be decomposed into two components: the mean force, which is constant along strike of the DOPB (centre column of four figures), and a remnant distribution of force (right column of four figures). Whereas the mean force contributes no net torque between component plates, the remnant force distribution contributes a net torque but no net force.

Figure 3

Sketch of distribution of forces in an idealized diffuse oceanic plate boundary (DOPB) when the pole of rotation lies along the central strike line. Two pole locations are are considered: (i) when the pole lies in the boundary and (ii) when it lies outside the boundary. Two end-member rheologies are also considered: (i) Newtonian viscous (corresponding to the n= 1 end-member of a power-law fluid) and (ii) plastic (yield strength) behaviour (corresponding to the n→∞ end-member of a power-law fluid). Arrows show the distribution of force as one moves along strike in the DOPB (left-hand column of four figures) for a flat Earth model. The distribution of forces can be decomposed into two components: the mean force, which is constant along strike of the DOPB (centre column of four figures), and a remnant distribution of force (right column of four figures). Whereas the mean force contributes no net torque between component plates, the remnant force distribution contributes a net torque but no net force.

In the two Newtonian cases, the length of the arrows are proportional to the deformation rate (as well as to the displacement rate, because the boundary is assumed to be of constant width) and therefore are proportional to the distance from the pole of rotation.

In the two plastic cases, the magnitude of force is simply equal to yield strength and therefore the arrows are of constant length. Thus, it is independent of the deformation rate (and of the displacement rate), but depends on the sign of the deformation and displacement with the force resisting contraction being in the opposite direction to the force resisting extension.

In each case, the force distribution can be decomposed into two parts, the mean force (which has no net torque about the centre of the boundary) and the remnant force (which has a net torque but no net force). The decompositions make it clear why the pole of rotation must lie inside the boundary for the plastic case. If the pole of rotation were to lie outside the boundary, the resulting deformation could produce no torque about the centre of the boundary. Thus, the pole could lie outside the boundary only if there is no component of externally applied torque about the centre of the boundary, the probability of which is vanishingly small. In contrast, for the Newtonian case, a pole of rotation outside the boundary can generally produce a torque about the centre of the boundary, but not as large a torque as when the pole is in the boundary (for the same rate of rotation).

Thus, it is clear that the two end-member rheologies behave qualitatively very differently if the pole of rotation lies outside the along-strike limits of the diffuse plate boundary.

## Model Setup, Assumptions And Methods

We assume that the deformation across the boundary takes place over some width W, which will generally be a function of θ. If the deformation can be approximated by a uniform contraction or extension across the width and if the boundary consists of a thin sheet of viscous fluid, one can define a vector rate of deformation function, d, along GC such that d=Δu/W= (1/W) ω×r. Δu is the velocity jump across the boundary calculated along GC and r is the position vector relative to the centre of the Earth. If one then assumes a power-law relation between d and the local increment of force, δf, along the boundary acting on the plate through an increment of boundary δl (=rδθ) along GC, then

1
where is a constant of proportionality and , where . (The tensor N arises in a thin viscous sheet because normal velocity gradients lead to twice the strain rate and hence twice the stress, τ, of shear velocity gradients, i.e. τii= 2η∂ui/∂xi, but τij=η∂ui/∂xj if ij and ∂uj/∂xi can be neglected.) We consider only the contributions from gradients of displacement with respect to ψ and neglect the effects of gradients with respect to θ. Although the gradients with respect to θ can be comparable in size to those with respect to ψ, they do not contribute significantly to the force or torque that one component plate exerts on the other (or vice versa).

One of these two gradients, ∂uθ/∂θ, can give rise to no force on a vertical plane parallel to the central strike line. This gradient corresponds to the elongation (or shortening) of the diffuse plate boundary in the boundary-parallel direction (i.e. tectonic escape or its inverse).

The second of these gradients, ∂uψ/∂θ, could potentially contribute to a force on a vertical plane parallel to the central strike line, as it contributes to the strain component εxy. This gradient corresponds to horizontal simple shearing on vertical surfaces perpendicular to the central strike line. Here, we analyse the kinematics in a frame of reference in which the two component plates have equal and opposite rotations (at rates half the total relative rotation of the two plates). In such a frame of reference, the sign of ∂uψ/∂θ changes from the +y side of the deforming zone to the −y side of the zone. In the special case in which the velocity field is symmetric about the central strike line, ∂uψ/∂θ has opposite signs on the two sides of the central strike line; uψ vanishes along the central strike line and thus ∂uψ/∂θ also vanishes along the central strike line. Thus, this gradient can be ignored in the calculation of the force and torque that one component plate exerts on the other if they are calculated along the central strike line.

This is illustrated by a simple planar kinematic model in which two component plates rotate in equal and opposite directions (Fig. 4). In this case, the plates are diverging about a pole of rotation (located at coordinates 0,0) along the central strike line (upper panel of Fig. 4). Black-filled arrows show the velocity fields of the bounding plates along the edges of the deforming zone. We require that the velocity in the deforming zone be continuous across the edges of the two bounding component plates. We seek a simple velocity field consistent with the boundary conditions. To do so, we linearly interpolate uy along lines of constant x, which results in values of ∂uy/∂y that do not vary with y and increase (in magnitude) linearly with x. Similarly, we interpolate ux along lines of constant x; because ux has the same value everywhere along the edges of the deforming zone, this results in a uniform value of ux everywhere in the deforming zone. Thus, both partial derivatives of ux vanish everywhere in the deforming zone (Fig. 4).

Figure 4

Upper: velocity field for simple kinematic model of distributed deformation across a diffuse oceanic plate boundary (DOPB). Filled arrows are the velocities of the edges of the bounding rigid plates. The x component of velocity, ux, is identical throughout the deforming zone. Thus, both ∂ux/∂x and ∂ux/∂y are identically zero in the deforming zone. The y component of velocity, uy is linearly interpolated between bounding rigid plates along lines of constant x. Lower: contours of components of the 2-D velocity gradient tensor. Solid contours: ∂uy/∂y in arbitrary units. The magnitude of this component of the velocity gradient tensor increases monotonically with increasing x. Dashed contours: ∂uy/∂x in the same arbitrary units. This component of the velocity gradient tensor changes sign with y and vanishes along the central strike line, where it is negligible relative to ∂uy/∂y (except for points arbitrarily close to the origin).

Figure 4

Upper: velocity field for simple kinematic model of distributed deformation across a diffuse oceanic plate boundary (DOPB). Filled arrows are the velocities of the edges of the bounding rigid plates. The x component of velocity, ux, is identical throughout the deforming zone. Thus, both ∂ux/∂x and ∂ux/∂y are identically zero in the deforming zone. The y component of velocity, uy is linearly interpolated between bounding rigid plates along lines of constant x. Lower: contours of components of the 2-D velocity gradient tensor. Solid contours: ∂uy/∂y in arbitrary units. The magnitude of this component of the velocity gradient tensor increases monotonically with increasing x. Dashed contours: ∂uy/∂x in the same arbitrary units. This component of the velocity gradient tensor changes sign with y and vanishes along the central strike line, where it is negligible relative to ∂uy/∂y (except for points arbitrarily close to the origin).

uy/∂x corresponds to strike-slip faulting on planes parallel to the y-axis. Values of ∂uy/∂x (Fig. 4) show that it is greatest adjacent to the edges of the bounding plates. Whereas left-lateral strike-slip faulting occurs in the upper (+y) half of the diagram, right-lateral strike-slip faulting occurs along the lower (−y) half of the diagram, as indicated by the change in sign of the gradient. Along the central strike line, except possibly near the origin, it is clear that ∂uy/∂x can be neglected relative to ∂uy/∂y.

The effect of shear on vertical planes perpendicular to the central strike line is to redistribute the force per unit length resulting longitudinal strain along lines of constant x. The main effect is to redistribute the force per unit length from regions of high longitudinal strain to adjacent regions of low longitudinal strain, making it even more difficult to balance a torque about the middle of a diffuse plate boundary. Thus, the key effect relevant here is that a better approximation than that we use would show that it is even more unlikely for the pole of rotation to lie outside the DOPB than we find below.

In any event, with the assumptions above it follows that

2
where the direction is parallel to on GC, and L is the boundary length. For a Newtonian boundary (n= 1), α=h|r|η where η is the viscosity and h is the lithospheric thickness.

Some analytical solutions of eq. (2) can be found for particular values of n or for limiting cases or both (Zatman et. al 2001). Here, we numerically integrate eq. (2) assuming that η and h are independent of r. We further assume that the DOPB has constant width W, which is consistent with interpretation of seismic profiles across the Central Indian basin (van Orman et. al 1995). Even if the boundary changes in width in some systematic way, the constant-width approximation may still be made useful by modifying the exponent in the power law. For example, if the boundary material is Newtonian and W is proportional to sine of the distance from the pole of rotation (i.e. ), then the deformation function d is constant and this case becomes identical to constant Wand the limit n→∞. In general, if the width increases with increasing distance from the pole of rotation, the dynamics of the boundary can be approximated by an exponent larger than that appropriate if the boundary has a constant width.

## Sensitivity To Rheology

One can think of the angular velocity between component plates as being a response of oceanic lithosphere to the torque that each component plate applies to the other. The numerical integrations allow us to evaluate quantitatively the sensitivity of that response to the vertically averaged rheology of deforming oceanic lithosphere. Here, we examine how rheology alters the relationship between rotation and torque.

Fig. 5 shows the ratio Tz/Tx, normalized to the Newtonian ratio, as a function of the power-law exponent for two cases. In the first case (dashed curve), for which the pole of rotation lies along the central strike line and inside the boundary, there is only a modest decrease, from 1.00 to 0.84 in normalized Tz/Tx between n= 1 and n→∞. Most of the change occurs for small values of the power-law exponent (Tz/Tx equals 0.88 for n= 3, 0.85 for n= 6 and 0.84 for n= 10). In contrast, when the pole of rotation lies along the central strike line but outside the boundary (solid curve), normalized Tz/Tx decreases from 1.00 to 0.00 between n= 1 and n→∞. Most of the change also occurs for small values of the power-law exponent (Tz/Tx equals 0.37 for n= 3, 0.19 for n= 6 and 0.04 for n= 10).

Figure 5

Ratio of torque about the z-axis to torque about the x-axis as a function of the power-law exponent n. This ratio is normalized to the ratio for the Newtonian case (n= 1). For this example, a plate boundary 30° in arc length was used. The dashed curve is for a pole of rotation lying 5° inside (i.e. 10° from the centre of) the diffuse oceanic plate boundary (DOPB), and the solid curve is for a pole of rotation lying 5° outside the DOPB.

Figure 5

Ratio of torque about the z-axis to torque about the x-axis as a function of the power-law exponent n. This ratio is normalized to the ratio for the Newtonian case (n= 1). For this example, a plate boundary 30° in arc length was used. The dashed curve is for a pole of rotation lying 5° inside (i.e. 10° from the centre of) the diffuse oceanic plate boundary (DOPB), and the solid curve is for a pole of rotation lying 5° outside the DOPB.

These results show that when the pole of rotation lies outside the boundary, the inferred torque is sensitive to the rheology but most of this sensitivity occurs for small power-law exponents with the largest part being between the Newtonian and n= 3 cases. On the other hand, when the pole of rotation lies inside the boundary, the resulting torque is much less sensitive to the rheology. The small dependence on the power-law exponent is concentrated in small exponents and almost all of this between the Newtonian and n= 3 cases. Thus, if the pole of rotation lies inside the boundary, the relation between the torque and pole of rotation is insensitive to the rheology.

Fig. 6 shows the distribution of forces in an idealized DOPB for the same two cases shown in Fig. 5 and for power-law fluids with n= 1, 3, 10, 30 and n→∞.

Figure 6

Left column: distribution of forces in an idealized diffuse oceanic plate boundary (DOPB) for five distinct power-law rheologies (long-short-short dashed, n= 1; dotted, n= 3, evenly dashed, n= 10; dot-dashed, n= 30; solid, n→∞). The distribution of forces is decomposed into two components: the mean force, which is constant along strike of the DOPB (centre column), and a remnant distribution of force (right column). The upper row of panels is for a pole of rotation lying 5° inside (i.e. 10° from the centre of) the DOPB and the lower row of panels is for a pole of rotation lying 5° outside the DOPB. In both rows, the pole lies along the central strike line and the arc length of the plate boundary is 30°.

Figure 6

Left column: distribution of forces in an idealized diffuse oceanic plate boundary (DOPB) for five distinct power-law rheologies (long-short-short dashed, n= 1; dotted, n= 3, evenly dashed, n= 10; dot-dashed, n= 30; solid, n→∞). The distribution of forces is decomposed into two components: the mean force, which is constant along strike of the DOPB (centre column), and a remnant distribution of force (right column). The upper row of panels is for a pole of rotation lying 5° inside (i.e. 10° from the centre of) the DOPB and the lower row of panels is for a pole of rotation lying 5° outside the DOPB. In both rows, the pole lies along the central strike line and the arc length of the plate boundary is 30°.

If the pole of rotation lies inside the boundary along the central strike line, the force changes sign across the pole of rotation and all the modelled rheologies produce a distribution of forces that can produce a torque capable of balancing a torque applied about the middle of the boundary (upper half of Fig. 6). Thus, if the pole of rotation lies in the boundary, the inferred orientation of the torque is insensitive to rheology.

If the pole of rotation lies outside the boundary along the central strike line, the force across the boundary nowhere changes sign and the distribution of forces is capable of making only a small contribution to the torque about the middle of the boundary except in the Newtonian case and perhaps in the n= 3 case (lower half of Fig. 6). If and a significant torque is applied about the middle of the boundary, the pole of rotation is therefore unlikely to lie outside the boundary in the along-strike direction. The results for power-law fluids with resemble the n→∞ case much more than they resemble the Newtonian case, with the n= 3 case lying about halfway between the two end-members.

Fig. 7 illustrates how unlikely it is that the pole of rotation lies outside the boundary for . Moreover, it shows that this result depends only weakly on rheology, especially if only power-law rheologies with n≥ 3 are considered. For an oceanic plate consisting of a brittle upper lithosphere over a lower lithosphere that deforms viscously via dislocation creep (for which n≃ 3), the overall effective n for the vertically averaged response should be between 3 and ∞ and probably less than ≈30 (Gordon 2000). There is little difference between the curves for n= 3, 6, 10 and n→∞ in any of the graphs (Fig. 7). Except when (Fig. 8), the values of ωθ given by these curves differ by less than 2° and are generally much closer (Fig. 7). Our numerical results thus show that one need not know the value of n exactly, because n= 3 and n→∞ are nearly indistinguishable.

Figure 7

Variations of the angular coordinates of the pole of relative motion (ωθ, ωψ) with the location of the boundary torque (TθTψ), the boundary length L and the power-law exponent n. Each graph contains curves for n= 1, 3, 6, 10 and n large (to approximate n→∞). The left column (panels a to c) shows how ωθ varies with Tθ when Tψψ= 0, the middle column (panels d to f) shows how ωθ varies with Tθ when Tψ= 30° and the right column (panels g to i) shows how ωψ varies with Tθ when Tψ= 30°. In the top row the boundary length is L= 15° (similar to the North America—South America diffuse boundary), in the middle row L= 30° (similar to the India—Capricorn and Capricorn—Australia diffuse boundaries) and in the bottom row L= 55° (similar to the Nubia—Somalia diffuse boundary). The main implication of these graphs is that the vertically averaged rheology makes little difference: the curves for n= 3, 6, 10 and n→∞ lie nearly on top of each other, and are similar to the curve for n= 1 in each graph. The horizontal lines in panels (a) to (f) are reference lines for ωθ. Whereas values of ωθ larger than the reference value indicate that the pole of rotation lies outside the diffuse oceanic plate boundary (DOPB; in the along-strike direction), values of ωθ smaller than the reference value indicate that the pole of rotation lies in the DOPB. Similarly, the shaded bands in panels (g) to (i) are reference values for ωψ. Whereas values of ωψ larger than the reference value indicate that the pole of rotation lies outside the DOPB (in the across-strike direction), values of ωψ smaller than the reference value indicate that the pole of rotation lies in the DOPB.

Figure 7

Variations of the angular coordinates of the pole of relative motion (ωθ, ωψ) with the location of the boundary torque (TθTψ), the boundary length L and the power-law exponent n. Each graph contains curves for n= 1, 3, 6, 10 and n large (to approximate n→∞). The left column (panels a to c) shows how ωθ varies with Tθ when Tψψ= 0, the middle column (panels d to f) shows how ωθ varies with Tθ when Tψ= 30° and the right column (panels g to i) shows how ωψ varies with Tθ when Tψ= 30°. In the top row the boundary length is L= 15° (similar to the North America—South America diffuse boundary), in the middle row L= 30° (similar to the India—Capricorn and Capricorn—Australia diffuse boundaries) and in the bottom row L= 55° (similar to the Nubia—Somalia diffuse boundary). The main implication of these graphs is that the vertically averaged rheology makes little difference: the curves for n= 3, 6, 10 and n→∞ lie nearly on top of each other, and are similar to the curve for n= 1 in each graph. The horizontal lines in panels (a) to (f) are reference lines for ωθ. Whereas values of ωθ larger than the reference value indicate that the pole of rotation lies outside the diffuse oceanic plate boundary (DOPB; in the along-strike direction), values of ωθ smaller than the reference value indicate that the pole of rotation lies in the DOPB. Similarly, the shaded bands in panels (g) to (i) are reference values for ωψ. Whereas values of ωψ larger than the reference value indicate that the pole of rotation lies outside the DOPB (in the across-strike direction), values of ωψ smaller than the reference value indicate that the pole of rotation lies in the DOPB.

Figure 8

The critical value of the torque angle in the along-strike direction θT at which the pole of rotation lies exactly on the edge of the boundary in that direction (i.e. ωθ=L/2), as a function of n (the power-law exponent), for different values of L and Tψ. Increasing n (which corresponds to a more plastic-like rheology) increases the probability that the pole of rotation will be confined to the diffuse oceanic plate boundary (DOPB) in the along-strike direction.

Figure 8

The critical value of the torque angle in the along-strike direction θT at which the pole of rotation lies exactly on the edge of the boundary in that direction (i.e. ωθ=L/2), as a function of n (the power-law exponent), for different values of L and Tψ. Increasing n (which corresponds to a more plastic-like rheology) increases the probability that the pole of rotation will be confined to the diffuse oceanic plate boundary (DOPB) in the along-strike direction.

Fig. 8 shows the locus of critical values of Tθ as a function of the power-law exponent, where a critical value of Tθ is one that causes the pole of rotation to lie along the edge of the DOPB. The figure illustrates several key results. First, it shows that the pole of rotation is more tightly confined to the boundary for shorter boundaries. Secondly, it shows that the the pole of rotation is more tightly confined to the boundary in the along-strike direction for torques along the central strike line than off the central strike line. Thirdly, Zatman et. al (2001) showed from analytical models that the pole of rotation in the diffuse boundary is better confined in the along-strike direction for n→∞ than for n= 1 when Tψ= 0. Fig. 8 shows that this is a special case of a more general rule that the pole is better confined in the along-strike direction as the power-law exponent increases. Fourthly, the figure shows that the dependence on the power-law exponent is weak for n≥ 3 and very weak for n≥ 10.

Fig. 9 displays maps of the critical orientation of torque that cause angular velocities precisely parallel to the edge of the diffuse plate boundary. It shows that the pole of rotation is more tightly confined, not only in the along-strike direction, but also in the across-strike direction as the power-law exponent increases. It reinforces the conclusions that the results are insensitive to the power-law exponent for n≥ 3 and that the pole is more strongly confined to the boundary for short boundaries than for long boundaries.

Figure 9

The critical value of the torque orientation at which the pole of rotation lies exactly on the edge of the diffuse oceanic plate boundary (DOPB) is shown for values of n= 1 (dotted curve), 3 (dashed curve), 6 (long-short dashed curve), 10 (long-short-short dashed curve) and large (n→∞, solid curve). Sets of curves are shown for three different lengths of DOPBs: 55° (similar to the Nubia—Somalia boundary), 30° (similar to the India—Capricorn and Capricorn—Australia boundaries) and 15° (similar to the North America—South America boundary). For a given power-law exponent and boundary length, a torque orientation lying inside the curve would give rise to a pole of rotation inside the DOPB, while a torque orientation lying outside the curve would give rise to a pole of rotation outside the DOPB. For all three boundary lengths, the curve for n= 3 lies closer to n= large than to n= 1 and the chances of the pole of rotation lying outside the boundary in the across-strike direction are much higher than the chances in the along-strike direction. As boundary length decreases, the fraction of possible torque orientations that gives rise to a pole of rotation inside the DOPB increases.

Figure 9

The critical value of the torque orientation at which the pole of rotation lies exactly on the edge of the diffuse oceanic plate boundary (DOPB) is shown for values of n= 1 (dotted curve), 3 (dashed curve), 6 (long-short dashed curve), 10 (long-short-short dashed curve) and large (n→∞, solid curve). Sets of curves are shown for three different lengths of DOPBs: 55° (similar to the Nubia—Somalia boundary), 30° (similar to the India—Capricorn and Capricorn—Australia boundaries) and 15° (similar to the North America—South America boundary). For a given power-law exponent and boundary length, a torque orientation lying inside the curve would give rise to a pole of rotation inside the DOPB, while a torque orientation lying outside the curve would give rise to a pole of rotation outside the DOPB. For all three boundary lengths, the curve for n= 3 lies closer to n= large than to n= 1 and the chances of the pole of rotation lying outside the boundary in the across-strike direction are much higher than the chances in the along-strike direction. As boundary length decreases, the fraction of possible torque orientations that gives rise to a pole of rotation inside the DOPB increases.

These relationships are further quantified in Fig. 10, which shows the percentage of solid angle enclosed by the curves of critical torque orientation for three different boundary lengths as a function of the power-law exponent. This figure also reinforces the inference that the results are insensitive to the value of the power-law exponent for n≥ 3 and especially for n≥ 10. The figure can be used to infer the probability that the pole of rotation lies in a DOPB. Consider the results if n= 10, which may be appropriate for mature oceanic lithosphere (Gordon 2000). If one assumes that all orientations of torque are equally likely, there is a 49 per cent chance of the pole of rotation lying outside the DOPB for a boundary length of 55°, a 21 per cent chance of the pole of rotation lying outside the boundary for a boundary length of 30° and a 6 per cent chance of the pole of rotation lying outside the boundary for a boundary length of 15°.

Figure 10

Per cent of torque orientations that result in poles of rotation inside a diffuse oceanic plate boundary (DOPB), found by integrating the area inside the curves of Fig. 9 as a function of the power-law exponent. The results reinforce the conclusion that most of the sensitivity to the power-law exponent is in the smallest exponents (n < 10 and especially n < 3). Moreover, the curves quantify the probability that the pole of rotation lies outside the boundary if the orientation of torque between component plates is uniformly random. Specifically, for all but the smallest exponents, the probability of lying outside the curve is 49 per cent if the boundary is 55° long (similar to the Nubia—Somalia boundary), 21 per cent if the boundary is 30° long (similar to the India—Capricorn and Capricorn—Australia boundaries) and 6 per cent if 15° long (similar to the North America—South America boundary).

Figure 10

Per cent of torque orientations that result in poles of rotation inside a diffuse oceanic plate boundary (DOPB), found by integrating the area inside the curves of Fig. 9 as a function of the power-law exponent. The results reinforce the conclusion that most of the sensitivity to the power-law exponent is in the smallest exponents (n < 10 and especially n < 3). Moreover, the curves quantify the probability that the pole of rotation lies outside the boundary if the orientation of torque between component plates is uniformly random. Specifically, for all but the smallest exponents, the probability of lying outside the curve is 49 per cent if the boundary is 55° long (similar to the Nubia—Somalia boundary), 21 per cent if the boundary is 30° long (similar to the India—Capricorn and Capricorn—Australia boundaries) and 6 per cent if 15° long (similar to the North America—South America boundary).

The diffuse boundaries between the (i) North American and South American plates, (ii) Indian and Capricorn plates, (iii) Capricorn and Australian plates, and (iv) Nubian and Somalian plates respectively have along-strike lengths of about 15°, 30°, 30° and 55°. In any event, the pole of rotation in the first two cases is known to lie within the respective diffuse plate boundaries (Royer & Gordon 1997; Gordon 1998), probably lies in the boundary in the third case (Royer & Gordon 1997; Conder & Forsyth 2001) and may either lie near to or far outside the boundary in the fourth case (Chu & Gordon 1999; Horner-Johnson et. al 2005), which has the highest probability of lying outside the boundary. Moreover, if it lies outside the boundary, it does so in the across-strike direction, which Fig. 9 shows is much more likely to occur than a pole lying outside the boundary in the along-strike direction.

The location of poles of rotation of component plates are known observationally only to within a few degrees at best (in the case of the Indian and Capricorn component plates) and to within tens of degrees at worst (Capricorn—Australia and Macquarie—Australia; Royer & Gordon 1997; Conder & Forsyth 2001; Cande & Stock 2004). Therefore the observed locations of the poles of rotation are insufficiently precise to be able to distinguish between the different rheological models, even if one knew the torques exactly. An additional benefit is that knowledge of the across-strike width of the boundary and the along-strike variation in width is no longer needed for estimating this orientation. If one assumes that n is large, then eq. (2) shows that variations in W are unimportant, so one ne not know the across-strike profile of the deformation. This is because n→∞ is the plastic limit, in which case the stress locally in the zone of deformation is the yield stress. Otherwise, for finite neq. (2) shows that the stress is proportional to W−1/n. Even for n= 3, doubling the width only decreases the stress by 21 per cent; for n= 10, doubling the width only decreases the stress by 7 per cent.

## Discussion

The most important result from our study is that the relationship between the orientation of the torque that one component plate exerts on another and the orientation of the angular velocity between the two plates is insensitive to the rheology. This lays the groundwork for future work. Our results imply that the distribution of stresses can be inferred from simply knowing the location of the boundary and the relative plate velocity across it. It is not necessary to know the material properties or width of the zone to infer much of the dynamics. This implies a relatively simple link between the kinematics and dynamics of DOPBs in which the orientation of the torque can be inferred from the orientation of the angular velocity.

On the other hand, the results presented here tell us little about the average strength of the lithosphere because the analysis is concerned only with the orientation and not the magnitude of the torque. It does, however, lay the framework for attempting to find such magnitudes by balancing torques across DOPBs with torques along other plate boundaries for which the magnitudes can be estimated. Similarly, our results tell us little about the value of the power-law exponent appropriate for vertically averaged deforming oceanic lithosphere. This can be estimated, however, from the shape of deformation profiles (England et. al 1985). If so, it would provide a means of further constraining how strength varies with depth in deforming oceanic lithosphere (Gordon 2000).

## Conclusions

The relationship between the angular velocity between two component plates having a mutual diffuse plate boundary, on the one hand, to the torque on each component plate from the boundary, on the other hand, depends little on the vertically averaged rheology of the diffuse plate boundary in so far as it can be parametrized as a power-law fluid. Consequently the relationship is also insensitive to the width of the boundary in the across-strike direction. Thus, our results imply that the distribution of stresses can be inferred from simply knowing the location of the boundary and the relative plate velocity across it. It is not necessary to know the material properties or width of the zone to infer much of the dynamics. This implies a relatively simple link between the kinematics and dynamics of DOPBs in which the orientation of the torque can be inferred from the orientation of the angular velocity.

### Acknowledgments

SZ was the lead worker on this research and it is fitting that he be first author. RGG has revised some of the equations from earlier drafts of the manuscript, however, and is responsible for any errors that remain (or were introduced). This work began when SZ and RGG were both at the University of California at Berkeley with support from the the Miller Institute for Basic Research in Science. RGG further acknowledges support from National Science Foundation grant OCE-0242904. We thank Mark Richards for helpful discussions on this topic, and Jean-Claude De Bremaecker and Adrian Lenardic for helpful suggestions for improvement of an earlier version of this manuscript. Benjamin Horner-Johnson helped with preparing some of the figures.

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