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 use analytical and numerical models to investigate the dynamics of diffuse oceanic plate boundaries assuming that the vertically averaged rheology of deforming oceanic lithosphere is characterized by a power-law rheology, that is, , where 1 ≤ n ≤ ∞, is strain rate, and τ is deviatoric stress. A major conclusion of this prior work is that the pole of relative rotation of plates adjoining a diffuse oceanic plate boundary is predicted to be confined to the diffuse oceanic plate boundary, especially if n ≥ 3.
Here we use laboratory experiments to test the general validity of this conclusion. In these analogue experiments, deforming oceanic lithosphere is modelled alternately with a Newtonian fluid (n = 1) and a power-law fluid (n ≈ 6–10). The experimental results are consistent with the analytical and numerical models. In particular, the experimental results confirm that for a given applied force the pole of rotation is more strongly confined to the laboratory analogue of a diffuse oceanic plate boundary for a power-law fluid than it is for a Newtonian fluid. For forces applied far enough from the centre of the analogue diffuse plate boundary, however, results for Newtonian and power-law fluids are similar.
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 most deformation is localized in these narrow boundaries, additional broad zones of deformation occur within traditionally defined ‘plates’. Displacement across these so-called ‘diffuse plate boundaries’ takes place at rates that are generally lower than displacement rates 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).
Diffuse oceanic plate boundaries have linear dimensions of 1000 km or more and include the boundary between the North American plate and South American component plates (Ball & Harrison 1970; Argus 1990; Gordon 1998), the boundaries that separate the Indo-Australian composite plate into the Indian, Capricorn, Australian and Macquarie component plates (Fig. 1) (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). Diffuse plate boundaries within such composite plates are termed ‘complete diffuse boundaries’ (Zatman et al. 2001) as they terminate at triple junctions, as opposed to ‘partial diffuse boundaries’, which transitions into narrow plate boundaries. 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).
The dynamics of diffuse oceanic plate boundaries are mainly driven by tectonic forces applied at the edges of component plates and possibly also by buoyancy forces associated with the cooling and subsidence with age of oceanic lithosphere. These driving forces are balanced by three restoring forces: viscous flow stresses arising in response to deformation of the boundary itself, mantle tractions on the base of the lithosphere and buoyancy stresses arising due to deformation-driven variations in lithospheric thickness. Several studies have shown, however, that oceanic plates are sufficiently decoupled from the mantle by the low-viscosity asthenosphere that the effect of mantle tractions may be neglected (England & McKenzie 1982; Richards et al. 2001). In addition, owing to the high effective viscosity inferred for deforming oceanic lithosphere (Martinod & Molnar 1995; Gordon 2000; Gerbault 2000), buoyancy-driven motions due to deformation-driven variations in the thickness of the oceanic lithosphere can also be neglected over the timescale of deformation in diffuse oceanic plate boundaries. Thus, to first order, the dynamics of diffuse oceanic plate boundaries are governed by a balance between plate-edge driving forces, buoyancy forces associated with cooling and subsidence with age of oceanic lithosphere, and retarding viscous forces, which depend on the rheology of the boundary itself.
Prior analytical and numerical models apply the further simplified balance between driving plate-edge forces and retarding viscous forces to address the question of why the pole of rotation tends to lie in a complete diffuse oceanic plate boundary (Zatman et al. 2001, 2005). In particular, this work identifies conditions under which the pole is confined to the diffuse plate boundary and investigates the extent to which such conditions are sensitive to rheology. In this work, the vertically averaged rheology of deforming oceanic lithosphere is approximated as a power-law fluid (DeBremaecker 1977), for which , where is the strain rate, τ is the deviatoric stress, and n is the power-law exponent. Applying this approximation, Zatman et al. (2001) construct analytical models of diffuse boundary dynamics for the Newtonian and plastic (i.e. yield stress) end-member rheologies (i.e. n = 1 and n→∞). Zatman et al. (2005) construct numerical models for intermediate values of n. These models show for all n that the pole of relative rotation of component plates will be oriented closer to the centre of the boundary than will the torque. Moreover, the pole of rotation is unlikely to lie outside their mutual diffuse boundary irrespective of rheology, especially if n ≥ 3.
All displacements in the fluid are perpendicular to the initial strike line of the diffuse oceanic plate boundary.
Shear on vertical planes perpendicular to the strike line can be neglected.
The diffuse oceanic plate boundary is of constant strike-perpendicular width.
The deformation is uniformly distributed in the direction perpendicular to the strike of the diffuse oceanic plate boundary.
To relax these constraints and to test the general validity of the theory, we conduct analogue experiments aimed at relating the style of deformation within diffuse plate boundaries to the forces imposed on the adjacent component plates. Using a series of experiments in which an indenter (the analogue to the edge of a component plate) is immersed in, or placed next to, a viscous Newtonian fluid (corn syrup) or a material obeying a high-exponent power law (kaolin clay), we verify these theoretical results for two important cases, one of which has properties similar to that expected for the vertically averaged rheology of deforming oceanic lithosphere.
Pole of Rotation for Diffuse Oceanic Plate Boundaries on A Flat Earth
Force distribution along a diffuse oceanic plate boundary: qualitative considerations
It is instructive to discuss why the pole of rotation is expected to be more strongly confined to the diffuse oceanic plate boundary if its vertically averaged rheology is that of a high-exponent power-law fluid than if its vertically averaged rheology were Newtonian. In Fig. 2, the entire along-strike length of a boundary is shown. Two different poles of rotation, one in the diffuse oceanic plate boundary, 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 the four combinations of two poles with two rheologies. The arrows represent the local contribution from a segment of the boundary to the force across the boundary.
In the two Newtonian cases, the length of the arrows is proportional to the deformation rate (as well as to the displacement rate, since the boundary is assumed to be of constant width) and, therefore, is 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 hence the also the sign of the 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 show 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, if the pole of rotation were outside the boundary, the unbalanced torque about the centre of the boundary would cause one or the other or both adjacent component plates to accelerate about the centre of the boundary until the pole of relative plate rotation did indeed lie inside the boundary.
In contrast, for the Newtonian case, a pole of rotation outside the boundary produces a torque about the centre of the boundary. Thus, the two end-member rheologies lead to qualitatively different behaviour if the pole of rotation lies outside the along-strike limits of the diffuse plate boundary (Zatman et al. 2005).
A flat-earth model provides some insights into the physics of diffuse oceanic plate boundaries and is readily testable by analogue laboratory models. The theoretical problem is defined in Fig. 3(a) and the experimental analogue is shown in Fig. 3(b). The position of the pole of rotation of component plate A relative to component plate B is defined to be (x0, y0), and we consider a central strike line for the boundary that follows a straight line along y = 0 from x = −L/2 to x = L/2. Dynamic equilibrium requires that the force and torque exerted on each component plate through the diffuse boundary balance the sum of the forces and torques from the other boundaries of the component plate. The balance can be represented by claiming that the boundary must provide a net force and torque on component plate B equivalent to that from some force F applied at some point along the central strike line of the diffuse oceanic plate boundary xF = (D, 0) (Fig. 3a). In our experiments (discussed below) this equivalent force is applied only at points in the diffuse boundary, but in principle the equivalent force can also be applied outside the boundary.Zatman et al. 2001). Thus, given the components Fx, Fy, the distance that the force is applied from the centre of the diffuse plate boundary, D, and the along-strike length of the diffuse plate boundary, L, the location of the pole of rotation, (x0, y0), can be obtained. Note that x0 is independent of both Fx and Fy and that x0 ∝ L2/D. Thus x0 is particularly sensitive to the length of a diffuse plate boundary and also to where the external force is applied along that boundary. In contrast, y0 depends not only on L and D, but also on the ratio Fx/Fy, that is, it depends on the orientation of F but not on its magnitude. In particular, if F is perpendicular to the strike line, as we will limit it to be in our experiments, then y0 vanishes. In this case, eq. (1) indicates that the pole of rotation will lie inside the diffuse boundary in the along-strike (x) direction (i.e. −L/2 < x0 < L/2) unless F is applied within the middle third of the diffuse boundary (i.e. −L/6 ≤D≤L/6). For the case for which Fy vanishes for a plastic rheology, Zatman et al. (2001) show that for the asymptotic case of n→∞ Zatman et al. (2005) to make appropriate predictions for x0.
Our experiments are aimed at testing the theoretical relationship between the torque acting across a diffuse plate boundary and the consequent position of the pole of rotation for an analogue indenter (i.e. the along-strike edge of a component plate) undergoing a displacement due to a force applied normal to its boundary. We investigate how the pole of rotation for a component plate varies as a function of the rheology of the boundary and where the force is applied along its boundary (i.e. resulting in different orientations of the applied torque). The theory is insensitive to the power-law exponent when n≥ 3 (Zatman et al. 2005). Hence, we can determine the validity of the theory over a range of rheologies by conducting experiments in which the deforming medium is either Newtonian (n = 1) or approximately plastic (n≥ 3).
We conduct our analogue experiments in a 20 × 40 × 2.5 cm high Pyrex baking dish sketched in Fig. 3b. Our analogue indenter is a 1.5 × 10 × 1.2 cm high bar of aluminium. Small holes are drilled into one side of the indenter at 1 cm intervals measured from the centre of the bar. Working fluids are 1–2-cm-high layers of either commercial corn syrup or a kaolin-based clay. The corn syrup has a Newtonian rheology and a strongly temperature-dependent viscosity, which for our experimental conditions is given approximately byZatman et al. (2001) in the Newtonian (n = 1) and plastic limits (n→∞). Moreover, the kaolin-based clay provides a way to address the result of Zatman et al. (2005) that the theory for a perfectly plastic material (n→∞) holds approximately for all n≥ 3 (cf.eq. 3). In addition, the vertically averaged rheology of the oceanic lithosphere can be characterized with n of order 10 (Gordon 2000). Hence, the kaolin clay also provides a good analogue for the natural case. Hereinafter we refer to corn syrup as ‘the Newtonian case’ and to kaolin clay as ‘the power-law case’. Finally, the theoretical arguments leading to eqs (1) and (3) assume that the dynamics of diffuse plate boundaries are uninfluenced by mantle tractions because of the lubricating effect of a low-viscosity asthenosphere. To satisfy this mechanical boundary condition in our experiments, prior to adding our working fluid we apply a thin layer of Pam® cooking oil to the floor and walls of the baking dish.
Experiments were performed by sequentially (centre to the right followed by centre to the left) fitting the tip of a long metal probe into each hole and pushing the bar very slowly over a short distance. The probe was perpendicular to the sidewall of the bar and the displacement of the indenter was recorded with a digital video camera. In a given experiment each sequence of displacements was repeated several times and in different orders so that the integrity of the approximately frictionless bottom boundary condition and the reproduceability of the measurements was ensured. The pole of rotation for each displacement was determined by carefully resolving the motion of the centre of mass of the indenter into components of rotation and translation.
Dimensionless parameters and scaling considerations
The mechanical properties of a thin viscous sheet are specified by two dimensionless parameters: the power-law exponent n, discussed above and the Argand number (modified from England & McKenzie (1982)),
Here Δρ is the relevant density contrast, g is gravitational acceleration, h is a characteristic length scale, typically the thickness of the lithosphere, μeff is an effective viscosity, which can depend on temperature, T, strain rate, , and deviatoric stress, σ and U is a characteristic velocity. In eq. (6), the numerator is a scale for the buoyancy stress associated with deformation-driven variations in layer thickness. The denominator is the viscous stress required to deform the material at a reference strain rate, here taken to be U/h. Ar can also be viewed as the ratio of the timescale for the horizontal advection of fluid due to the motion of the indenter to the timescale over which resultant topography is relaxed through buoyancy-driven fluid flow. When Ar≥ 1, the gravitationally driven spreading of topography formed in front of the indenter acts as an important restoring force. Comparison of forward models of thin viscous sheet deformation with continental crustal thickness and indicators of deformation in the India-Eurasia (continent-continent) collision indicate that Ar≲ 1 if n = 3 (England & Houseman 1986).
Ar for deformation of oceanic lithosphere is expected to be substantially smaller. The strain rates in deforming oceanic lithosphere are at least an order of magnitude less than those in the deforming continental lithosphere of the India-Eurasia collision zone, which implies that the effective viscosity of vertically averaged oceanic lithosphere is at least an order of magnitude higher than that of vertically averaged deforming continental lithosphere (Gordon 2000). Moreover, oceanic lithosphere lacks the thick buoyant crust of continental lithosphere. The average thickness of continental crust is 38 km (Mooney et al. 1998) and that of oceanic crust is 7 km (White et al. 1992). Typical values assumed for the density of continental crust, oceanic crust and mantle, respectively, are 2700 kg m−3, 2900 kg m−3 and 3300 kg m−3 (Fowler 2005). These values give density contrasts between continental crust and mantle of 600 kg m−3 and between oceanic crust and mantle of 400 kg m−3. The buoyancy produced by horizontal crustal shortening is proportional to both the initial thickness of the crust and its density contrast with the mantle. Thus, buoyancy forces due to horizontal shortening of oceanic lithosphere are about an order of magnitude smaller than those for continental lithosphere. Taken together these comparisons indicate that the Argand number appropriate for deforming oceanic lithosphere is two or more orders of magnitude smaller than that appropriate for the deforming continental lithosphere of the India-Eurasia collision zone. Thus, a plausible Argand number to be applied in our experiments is ≲ 10−2.
In our experiments, the kaolin clay has a finite yield strength that exceeds the buoyancy forces arising due to its deformation and, hence, ensures a small-Ar condition. To obtain similarly small-Ar conditions in corn syrup we took advantage of its strongly temperature-dependent viscosity and chilled the syrup to around −5oC, achieving a viscosity μ > 105 Pa-s. For typical imposed velocities, U < 10−3 m s−1, and a density contrast, Δρ = 1430 kg m−3, Ar < 10−3, which is consistent with conditions anticipated for oceanic lithosphere. We note also that under these conditions, motions in the Newtonian case are always in the Stokes limit (i.e. the Reynolds number, Re < 10−7 and flows are reversible).
Another dimensionless parameter that enters this problem is the map-view aspect ratio of the indenter, A = L/ℓ = 6.7, which is held constant. Zatman et al. (2001) analyse the force distribution along a strike line and thus implicitly assume that A→∞, which is appropriate for the Earth, but not for our experiments. Although our indenter has a large aspect ratio, the influence of a finite width is small but non-negligible, and is analysed quantitatively below.
Figs 4 and 5 are photographs showing the initial and final positions of the indenter in three Newtonian and three power-law experiments. The leftmost panel for both working fluids is the control experiment in which the indenter is forced approximately at its centre. The centre and right panels show how the displacement varies as the force is applied to the left (power-law case) and right (Newtonian case) of the centre. As expected in both cases, when the force is applied very near the centre of the indenter, the displacement is mainly translation. The centre panels show that if the force is applied only a small distance from the centre of the indenter, the net displacement of the bar remains mostly by translation but there is a small rotational component due to a torque about an axis that is well off the indenter. Theoretically, it is expected that if our kaolin clay was perfectly plastic (i.e. as n→∞), rotation of the indenter is possible only if the pole of rotation lies within the indenter itself (Fig. 1). The proximity of the pole to the indenter is a consequence of a finite n, as shown in the next section. The rightmost panel shows the position at which the applied force causes a displacement resulting from a torque about a pole of rotation at the edge of the indenter. Comparison of these three panels illustrates that the position at which the applied force causes rotation about a pole in the indenter is sensitive to the rheology of the deforming fluid. In particular, the pole becomes confined in the indenter for force applied 50 per cent closer to the centre for the power-law case than for the Newtonian case.
A pertinent observation from the experiments is that the strains in the fluids caused by the displacements from the indenter were far from uniform in the fluid, in contradiction of one of the assumptions of Zatman et al. (2001, 2005). Instead, the strains were highest immediately adjacent to the indenter and decayed with distance from the indenter, qualitatively resembling results from numerical models for deformation of a thin viscous sheet (England & McKenzie 1982; England et al. 1985; England & Houseman 1986). This spatial variation in the strain field reinforces the importance of using our experiments to quantitatively check the accuracy of the predictions of Zatman et al. (2001, 2005) as well as the general validity of these models.
In Fig. 6, we compare experimental measurements of the position of the pole of rotation obtained for all experiments with theoretical predictions for the Newtonian (n = 1) and perfectly plastic (n→∞) cases (eqs 1 and 3). Following Zatman et al. (2005) we also show numerical calculations for intermediate rheological cases, characterized by a power-law exponent n with 1 < n < ∞. Let P = x0/L be the non-dimensionalized position of the pole of rotation and D′ = D/L be the non-dimensionalized position of the applied force, where L is the indenter length. For the Newtonian case measurements are in good agreement with the theoretical predictions. In particular, the pole of rotation lies within the indenter for D′ ≥ 0.2, outside the indenter for D′ < 0.2, and far from the indenter for D′ < 0.05, as predicted. We note, however, that for D′ > 0.25 measurements of P are approximately 20–50 per cent larger than expected from the theory. This discrepancy is due to the finite width ℓ of the indenter and is analysed below.
Measurements of P for the power-law case are in excellent agreement with the theoretical predictions for power-law exponents n≥ 3 when D′ > 0.05. Specifically, the pole of rotation lies within the indenter for D′≥ 0.1, outside the indenter for smaller D with a maximum P of 0.6 to 0.8 as D′→ 0. Measurements of P at D′→ 0 are consistent with predictions within uncertainties. This discrepancy is discussed in more detail below.
Discrepancies between measurements and theory
In the Newtonian case, the observed the pole of rotation is farther from the centre of the indenter than predicted for D′ > 0.25. This discrepancy cannot be explained by invoking a power-law rheology for the corn syrup since all the power-law rheologies predict similar values of P for D′≥ 0.2. We note, however, that the predictions of Zatman et al. (2001) were not intended to account for interactions of the fluid with the ends of an indenter with a finite width. Accordingly, when we modify the Newtonian theory (Appendix A) to account for these additional viscous stresses, larger values of P are predicted and a better fit to the data is recovered for D′ > 0.25, as shown in Fig. 6.
In the power-law case, measurements of P at D′ = 0.025 are overpredicted by the theory for n = 3 and underpredicted by the theory for the high-exponent power laws (n≥ 30). As indicated by the error bars, some of this discrepancy is due to uncertainties over the precise positioning of holes near the centre of the indenter. As with the Newtonian case, if frictional effects due to the finite width of the indenter are important, P would tend to lie closer to, not farther from, the centre of the indenter. However, in the experiments shown in Fig. 5 the indenter ends were essentially not in contact with the kaolin clay. In additional experiments in which the indenter ends were immersed they became quickly lubricated by the seepage of pore water from the clay to the clay/indenter interface. This seepage occurred as a result of shear between the indenter end and the clay during individual experiments and was enhanced if experiments were repeated. Water seepage did not influence the rheology of the clay in front of the indenter. For the small strains in our experiments, the kaolin clay in front of the indenter was deformed by the normal component of the viscous stress retarding motion of the indenter. Indeed experiments were repeated many times, in varying sequences and independently, and our measurements of P were consistently reproducible.
When D′ < 0.2, the pole of rotation is more strongly confined in the indenter for n > 1. In contrast, when D′≥ 0.15, the results are similar for the two fluids: For each value of D′ the average value of P < 0.5, indicating that the pole of rotation lies within the indenter, as expected from theory. The good agreement between the Newtonian and power-law rheologies in this regime shows that the dynamics are insensitive to rheology for forces applied at a sufficiently large distance from the midpoint of the indenter.
Spherical earth interpretation
Our experiments reinforce another conclusion of the work of Zatman et al. (2001, 2005), that the smaller the along-strike length of a diffuse oceanic plate boundary, the less likely is the pole of rotation to lie outside the boundary. This follows if one thinks of the indenter of our experiment as being along the edge of a diffuse boundary on a spherical earth instead of being on a flat earth, as assumed in most of this paper. Insofar as the force applied to the indenter is perpendicular to the strike of the indenter, as in our experiments, the corresponding torque on a spherical earth is located precisely 90° away from the point where the force was applied along the great circle tangent to the strike of the indenter. Such a great circle has a length of about 40000 km along Earth's surface. Within that great circle of length 40000 km, the only orientations of torque that result in poles of rotation outside the analogue diffuse plate boundary lie along about 6 cm in the Newtonian case (1.5 cm on either side of ±90° from the middle of the edge of the indenter) and about 2 cm in the power-law case, in other words along about 1 part in 106 of the great circle. If torques applied to component plates have a uniform and random distribution then the probability of a torque resulting in a pole of rotation outside a boundary as short as in our experiment is small indeed.
Concluding Remarks and Directions for Future Research
The results from our laboratory experiments support and generalize the prior conclusion that the pole of relative rotation of adjoining component plates will be more strongly confined to a diffuse plate boundary if the boundary has a power law rather than a Newtonian rheology. In addition, if a given external force is applied sufficiently far from the centre of a diffuse plate boundary (i.e. D′≥ 0.15) our experiments and the accompanying theory show that the behaviour of the boundary is similar, whether the rheology is Newtonian or characterized by a power law, where n≥ 3. Finally, our work reinforces the earlier conclusion that the pole of rotation is increasingly likely to lie in the diffuse plate boundary as the along-strike length of the boundary decreases.
Our work shows that the simple models of (Zatman et al. 2001, 2005) captured the essential behaviour of diffuse oceanic plate boundaries during deformation. Taken together, our experiments and theoretical models support a more general conclusion which is that the thin viscous sheet approximation to the dynamics of both continental and oceanic diffuse plate boundaries is reasonable. This inference suggests a number of directions for future research in which combined experimental, analytical and numerical studies may lead to significant improvements over our current understanding. In particular, measurements of the deformation field around analogue indenters would provide an important check on models of deformation of diffuse oceanic plate boundaries (Zatman et al. 2001, 2005) and of thin viscous sheets used to represent continental deforming zones (England & McKenzie 1982; England et al. 1985; England & Houseman 1986).
All three authors were initially partly or fully supported by the Adolphe and Mary Sprague Miller Institute for Basic Research in Science. Additionally, RGG's contributions were supported by NSF Grant OCE-0242904 and AMJ's contributions were supported by NSERC and the Canadian Institute for Advanced Research. This work is dedicated to the memory of Stephen Zatman, who inspired us to laugh and to wonder.
Appendix A:Effect of a Finite Indenter Width on N = 1 Case
Although the indenter in our experiments is much longer than it is wide, viscous drag along the bar ends presents an additional and non-negligible retarding force to motions in the y-direction. The force balance in the x-direction is unchanged from the flat-earth case investigated by Zatman et al. (2001):eq. 2 of Zatman et al. 2001) becomes Zatman et al. (2001) this condition leads to an expression for xo modified from eq. (1) eq. (1)