Download Experimental tests of simple models for the dynamics

Geophys. J. Int. (2006) 164, 624–632
doi: 10.1111/j.1365-246X.2005.02840.x
Experimental tests of simple models for the dynamics of diffuse
oceanic plate boundaries
A. Mark Jellinek,1 Richard G. Gordon2 and Stephen Zatman3, ∗
1 Department
of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4. E-mail: [email protected]
of Earth Science, Rice University, Houston, TX 77005 USA
3 Department of Earth and Planetary Sciences, Washington University, Saint Louis, MO 63130 USA
2 Department
GJI Tectonics and geodynamics
Accepted 2005 October 11. Received 2005 March 17
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 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, ˙ ∝ τ n , 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.
Key words: anolog experiments, diffuse plate boundaries, rheology of the lithosphere.
1 I N T RO D U C T I O N
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.
∗ Deceased.
624
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
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Journal compilation Experimental tests of simple models for dynamics of diffuse oceanic plate boundaries
30˚N
EURASIA
ARABIA
PH
PACIFIC
CR
SOMALIA
CIB
90ER
0˚
WB
CIR
INDO - AUSTRALIA
. RTJ
30˚S
IR
SW
SE
IR
60˚S
ANTARCTICA
A
60˚E
90˚E
120˚E
150˚E
180˚
INDIA
0˚
CAP
30˚S
AUSTRALIA
60˚S
60˚E
B
ANTARCTICA
MACQUARIE
90˚E
120˚E
150˚E
625
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
˙ ∝ τ n , 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.
To obtain these analytical and numerical results, Zatman et al.
(2001) and Zatman et al. (2005) made significant assumptions,
including:
(1) All displacements in the fluid are perpendicular to the initial
strike line of the diffuse oceanic plate boundary.
(2) Shear on vertical planes perpendicular to the strike line can
be neglected.
(3) The diffuse oceanic plate boundary is of constant strikeperpendicular width.
(4) The deformation is uniformly distributed in the direction perpendicular to the strike of the diffuse oceanic plate boundary.
Figure 1. Deformation of the Indian Ocean basin. (a) The Indo-Australia
plate defined in the traditional way with narrow plate boundaries shown with
heavy lines. Filled circles concentrated on the boundaries indicate earthquakes of magnitude 5 or greater. (b) The Indo-Australian composite plate
and some of its components and diffuse plate boundaries. In both diffuse
oceanic plate boundaries shown the pole of rotation (filled circles) lies approximately in the middle of the plate boundary, separating a part of the plate
undergoing horizontal shortening (black arrows) from a part of the plate undergoing horizontal extension (white arrows). NUVEL-1 plate boundaries
are also shown. Observed deformation between the India and Capricorn
component plates has the apparent form of a rigid indenter (the Indian component plate) impinging on a fluidized zone on the edge of the Capricorn
plate: the amplitude of the deformation away from the boundary falls off
abruptly on the Indian side and gradually on the Capricorn side (Gordon
2000). (Figure by J.-Y. Royer; From Gordon & Royer (2005)). Notation:
PH, Phillipine sea plate; CR, Carlsberg Ridge; CIR, Central Indian Ridge;
90ER, Ninetyeast Ridge; WB, Wharton basin; RTJ, Rodrigues triple junction; SWIR, Southwest Indian Ridge; SEIR, Southeast Indian Ridge; CAP,
Capricorn component plate.
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.
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
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
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Journal compilation 2 P O L E O F R O T AT I O N F O R D I F F U S E
O C E A N I C P L AT E B O U N D A R I E S
O N A F L AT E A RT H
2.1 Force distribution along a diffuse oceanic plate
boundary: qualitative considerations
626
A. M. Jellinek, R. G. Gordon and S. Zatman
Remnant Force Distribution
(No Net Force)
Mean Force (No Net Torque)
Force Distribution
Pole outside of boundary
Pole in boundary
Newtonian (n=1)
=
+
=
+
Plastic (n → ∞ )
Force Distribution
Remnant Force Distribution
(No Net Force)
Pole in boundary
=
+
Pole outside of boundary
Mean Force (No Net Torque)
=
+
Figure 2. 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 considered: (1) when the pole lies in the boundary and (2) when it lies outside the boundary. Two end-member rheologies are also
considered: (1) Newtonian viscous (corresponding to the n = 1 end-member of a power-law fluid) and (2) 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 side 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. From Zatman et al. (2005).
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.
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Journal compilation Experimental tests of simple models for dynamics of diffuse oceanic plate boundaries
Central
strike line
Diffuse
boundary
a.)
(acting on
F component
Pole of
rotation
(A wrt B)
Co
m
Pl p o
at n e
e A nt
plate B)
n
e B ent
Pl
at
po
Co
0
0
x
m
L/
2
L
D
(0,0)
y
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).
627
x
y
2.2 Theory
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 (x 0 , y 0 ), 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
x F = (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.
Neglecting the dependence, if any, of the equivalent force F and
distance D on the deformation along the diffuse boundary, it follows
for the Newtonian case that
L2
,
12D
(1)
Fx L 2
,
Fy 12D
(2)
x0 = −
y0 =
(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, (x 0 , y 0 ), can be obtained.
Note that x 0 is independent of both Fx and Fy and that x 0 ∝ L 2 /D.
Thus x 0 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, y 0 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 y 0 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 < x 0 < 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
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2006 The Authors, GJI, 164, 624–632
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Journal compilation b.)
y
Indentor
x
L
Corn Syrup
or
Kaolin Clay
Pam
L
h
Figure 3. (a) The problem definition of Zatman et al. (2001) for theoretical
investigation of diffuse oceanic plate boundaries on a flat earth. (b) The
corresponding experimental setup used in this study.
plastic rheology, Zatman et al. (2001) show that for the asymptotic
case of n → ∞
L2
x0 = D ± D 2 +
.
(3)
4
Here, because the pole of rotation is confined to the boundary if D is
positive, the negative root should be taken, and if D is negative, the
positive root should be taken. For intermediate n (i.e. n = 3, n = 6,
n = 10, n = 100, and n = 10 000), we apply the numerical methods
of Zatman et al. (2005) to make appropriate predictions for x 0 .
3 EXPERIMENTS
3.1 Overview
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
628
A. M. Jellinek, R. G. Gordon and S. Zatman
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).
3.2 Experimental setup
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 by
μ ≈ μc exp(−γ T ),
(4)
where μ c is a reference viscosity and the characteristic rheological
ln μ
temperature scale, γ = −( ddT
) ≈ 0.2. In contrast, the kaolin clay
has a power-law rheology that can be described approximately by:
˙ ∝ τ n ,
(5)
where ˙ and τ are the rate of strain and stress, respectively, and n
is a power-law exponent. For the small strains and strain rates characteristic of our experiments n is constrained experimentally to be
in the range 6–10. These fluids are chosen to test the general behaviour of the analytical models developed in Zatman 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.
3.3 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)),
Ar =
ρgh
.
μeff U/ h
(6)
Here ρ is the relevant density contrast, g is gravitational acceleration, h is a characteristic length scale, typically the thickness
of the lithosphere, μ eff 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
−2
experiments is <
∼10 .
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 smallAr conditions in corn syrup we took advantage of its strongly
temperature-dependent viscosity and chilled the syrup to around
−5o C, achieving a viscosity μ > 105 Pa-s. For typical imposed velocities, U < 10−3 m s−1 , and a density contrast, ρ = 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/ = 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
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Journal compilation Experimental tests of simple models for dynamics of diffuse oceanic plate boundaries
629
Figure 4. Initial (top) and final (bottom) images of experiments with corn syrup (n = 1) as the deforming medium. Solid and dashed lines indicate displacement
of the indenter. Filled white circles are the poles of rotation calculated for each case. See text for discussion.
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.
4 R E S U LT S
4.1 Qualitative results
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
Figure 5. Initial (top) and final (bottom) images of experiments with kaolin clay as the deforming medium. Solid and dashed lines indicate displacement of
the indenter. Filled white circles are the poles of rotation calculated for each case. See text for discussion.
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630
4.3 Discrepancies between measurements and theory
Distance of pole from indenter center, P = x0 / L
4
3.5
3
2.5
n≈∞
n=1
n=1*
2
n=102
1.5
n=3
1
n=6
n=30
0.5
0
0.05
Pole Outside
Pole Inside
n=10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Distance of applied force from indenter center, D' = D / L
Figure 6. Experimental results showing the relationship between the distance of the applied force from the centre of the indenter and the distance of
the resultant pole of rotation from the centre of the indenter for Newtonian
(open squares) and power-law exponent (open circles) fluids. Also shown
are theoretical predictions for fluids with a variety of power-law exponents.
Dashed curves are analytical approximations (eqs 1 and 3) for n = 1 (i.e.
Newtonian) and n → ∞ (perfectly plastic) (Zatman et al. 2001). Solid curves
are numerical solutions for intermediate values of n (Zatman et al. 2005).
Dotted curve is the result from eq. (1) modified to include the effects of
coupling of viscous stress due to coupling at the ends of the indenter (see
Appendix A). See text for discussion.
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 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.
5 DISCUSSION
4.2 Quantitative results
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 =
x 0 /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
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.
5.1 Experiments
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.
5.2 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 40 000 km along Earth’s surface. Within that great
circle of length 40 000 km, the only orientations of torque that result
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Journal compilation Experimental tests of simple models for dynamics of diffuse oceanic plate boundaries
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.
6 C O N C LU D I N G R E M A R K S A N D
D I R E C T I O N S F O R F U T U R E R E S E A RC H
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).
AC K N OW L E D G M E N T S
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.
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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):
L/2
Fx = αω
yo d x = αωyo L .
(7)
−L/2
Here, α is a constant and ω is a rotation rate. Incorporating the effect
of viscous shear stresses integrated over the width, , of the bar, the
force balance in the y-direction (cf. eq. 2 of Zatman et al. 2001)
becomes
L/2
Fy = αω
yo d x = −2αωxo (L + 2).
(8)
−L/2
At static equilibrium the sum of the moments about the origin must
vanish. Following Zatman et al. (2001) this condition leads to an
expression for xo modified from eq. (1)
xo = L2
,
12D
(9)
where the correction to eq. (1)
=
(L + 6)
.
L + 2
(10)
For our experimental system, = 1.46.
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2006 The Authors, GJI, 164, 624–632
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