A key element of plate tectonics on Earth is that the lithosphere is subducting into the mantle. Subduction results from forces that bend and pull the lithosphere into the interior of the Earth. Once subducted, lithospheric slabs are further modified by dynamic forces in the mantle, and their sinking is inhibited by the increase in viscosity of the lower mantle. These forces are resisted by the material strength of the lithosphere. Using geodynamic models, we investigate several subduction models, wherein we control material strength by setting a maximum viscosity for the surface plates and the subducted slabs independently. We find that models characterized by a dichotomy of lithosphere strengths produce a spectrum of results that are comparable to interpretations of observations of subduction on Earth. These models have strong lithospheric plates at the surface, which promotes Earth-like single-sided subduction. At the same time, these models have weakened lithospheric subducted slabs which can more easily bend to either lie flat or fold into a slab pile atop the lower mantle, reproducing the spectrum of slab morphologies that have been interpreted from images of seismic tomography.

A key element of plate tectonics is the recycling of lithospheric plates into
the mantle. Colder and more dense slabs, already having been subducted,
generate the driving force that pulls and bends tectonic plates below the
surface. Coupling of tectonic plates at the surface, shear stress on the
subducted plate, induction of mantle flow around the subducting plate, the
contact of slabs with upper mantle/lower mantle boundary, and the material
strength of plates generate resisting forces which resist bending inhibit
subduction. The particular form and speed of subduction is controlled by the
balance of the opposing forces

On Earth this process is asymmetric, occurring at convergent boundaries where one of the two plates is subducted, while the other plate, the overriding plate, remains at the surface. The mechanical strength of lithospheric material needs to be sufficiently strong so that the overriding plate can resist bending while being weak enough so that the subducting plate can bend and subduct.

Subducted slabs moving through the mantle and encountering the higher
viscosity lower mantle experience forces that deform the slab. These forces
are transmitted to the surface, the slab acting as a stress guide, affecting
the state of stress at the trench and the coupling of subducting and
overriding plates

Models produce a diverse set of mantle convection styles

Parameters common to all models.

Diagram of the common parameter space, initial, and boundary conditions for models.

We develop two-dimensional models of convective systems using the finite
volume code StagYY

Dimensional parameters are used, and parameters common to all models are given
in Table

We use a strongly temperature-dependent Arrhenius relation to calculate the
viscosity of a silicate mantle, shown in Eq. (

Material is subject to failure through an applied yield strength failure
criterion. The yield stress,

Weakening of the subducted slab is accomplished by lowering the maximum
viscosity at depth for material below 1000 K. In the models where
slab weakening is used, material 10 km below the depth of the
lithosphere/asthenosphere boundary is assigned a lower maximum viscosity than
the surface plates. We use plates of 50 and 80 km in this study, and the
weakening is applied to material below 60 and 90 km respectively. We
examine cases where the weakening is 1, 10, or 100 % (no weakening) of
the maximum viscosity cutoff of the surface plates. In a study by

Piled slab descending into the lower mantle.

A temperature field and particle cloud are the initial conditions for all
models. The temperature field is defined by a uniform background temperature
of 1600 K and plates with a half-space cooling temperature gradient. The
viscosity of the background and plates is calculated using
Eq. (

The surface of each model is comprised of two plates. The initial downgoing plate begins 20 km from the left wall. It extends to the center of the box (the initial trench) and then follows a path with radius of curvature 400 km at the surface of the slab into the mantle to a depth of 200 km.

The overriding plate occupies the right side of the box, from 20 km right of the trench to 200 km short of the right wall. The boundary conditions reflect along the sides of the model domain, free slip at the top and no slip along the bottom.

These models use a pseudo-free surface in the form of a low density layer at
the top of the model called sticky air

To promote one-sided subduction

Radius of curvature has been used in several studies as a metric of
subduction

The “spline” method has been used previously in studies to calculate a
minimum radius of curvature for subducting slabs in both numerical models and
in the Earth

In a second method, which we call the “angle” method, we first fit a curve
to the base of the crust. We then select crust particles from the trench to
the point where that curve has a slope of

Model parameters.

We ran 36 models, varying the plate strength and weakening parameter of slabs
(full list of parameters in Table

For models with a lower mantle, the slab descends through the upper mantle
and encounters the boundary between the upper and lower mantle. The descent
of the slab is inhibited at this boundary. Lateral motion of the slab tip is
also inhibited, either because the tip penetrates into the lower mantle, or
because the slab rests on the boundary and is subject to shear traction
between the slab and lower mantle. The slab then lies flat on the upper/lower
mantle boundary, after which it buckles and piles. Eventually the piled slab
descends through the boundary as a unit; see Fig.

For models with no lower mantle, the slab descends until it encounters the bottom of the model box. The bottom has a no-slip boundary condition. When the slab encounters the bottom of the box, it lies down and buckles in a manner similar to those models with a lower mantle in which the slab becomes embedded.

A comparison of the radius of curvature (RoC) calculated for each of the methods tested for model 14.

Radius of curvature over time for each method tested for model 14.

We calculated the radius of curvature for every time step and every model using the three methods described above. The initial condition imposed a radius of curvature of 400 km. In the 0th time step, no method returned 400 km. The fact that no method returns exactly the prescribed radius of curvature is a result of three factors: first, the 400 km radius of curvature used in the initial condition is at the surface of the plate, while the fit circles are mid-crust; second, the least squares fit to a cloud of randomly placed points is a statistical solution subject to noise; third, and most significant, no method selects all of the points from the trench to the tip of the slab. This angle method, in the case of a perfect circle, will select points down to a depth of 117 km. The depth method is prescribed to select points no deeper than 150 km. The spline method is also a statistical method, subject to an arbitrary smoothing parameter, and the radius of curvature at any given point is only sensitive to nearby particles. The depth method always returned the closest radius of curvature. The average across models for the 0th time step for depth, angle, and spline methods was 370, 337, and 113 km, respectively.

The radius of curvature (RoC) over time for models 18 and 36, each with the same parameters, except for the presence of a lower mantle. The inset shows the 1550 K isotherm for each of the models at 4.4 Ma. The tip of the slab in model 18 (with a lower mantle) has descended to a depth of 625 km, while in the same time the tip of the slab in model 36 (no lower mantle) is at a depth of 1180 km.

The change in radius of curvature over time for three models of different plate strength, models 11, 14, and 17. The inset shows the 1550 K isotherm at the plate interface.

The three methods of calculating the radius of curvature for subducted slabs
return differing values for any given time step in the model.
Figure

Figure

Figure

The change in radius of curvature for three models with the same plate strength and three different weakening parameters, models 15, 14, and 13. Bold lines show the period of slab buckling. The inset shows the radius of curvature with the plot shifted to the onset of buckling.

Figure

In Figs.

Figure

A similar comparison of models of equal plate strength and varied slab
strength is shown in Fig.

In Figs.

Figure

The observed states of models, relaxing from the initial condition, encountering the upper/lower mantle boundary and buckling, are associated with changes to the radius of curvature as calculated by the two circle methods. The spline method for calculating radius of curvature is not sensitive to these states. The depth method captures a greater number of crust particles than the angle method and is consequently less noisy. Noisier still is the spline method. The spline method is subject to an arbitrary smoothing penalty. The minimum radius of curvature is a function of far fewer points than either of the circle methods. The points closest to the point of minimum radius of curvature are fit smoothly and have the largest effect on the radius of curvature, while points at increasing distance have less weight.

1550 K isotherm for models 18 (left/orange), 17 (center/green), and
16 (right/blue). Surface plates are 80 km thick and have a maximum viscosity
of

Radii of curvature are commonly used to calculate bending dissipation which
is balanced against the gravitational potential energy of sinking slabs

The steeply dipping slabs apply a stronger torque on the overriding plate
than shallow dipping slabs. Experiments, numerical and analog, suggest that
tectonic plates are strong, with viscosity contrasts several hundred times
that of the mantle

550 K isotherm for models 3 (left/orange), 2 (center/green), and 1
(right/blue). Surface plates are 50 km thick and have a maximum viscosity
of

Numerical models and observation reveal a rich variety of subducted slab
morphologies. Computational models may modify plate strength and rheological
laws to a variety of subduction styles

Other work has suggested that slabs stalled at the upper/lower mantle
boundary may eventually “avalanche” into the lower mantle

In this study the weakening mechanism is active at a depth below the base of
the surface plates. The range of weakening was selected so that we could
explore the effects of reduced viscosity on the subducted lithosphere. Work
done by

1550 K isotherm for models 12 (left/orange), 14 (center/green), and
16 (right/blue). Surface plates are 80 km thick and have a maximum
viscosity of

Though the weakening mechanism is turned on at these shallow depths, the
morphology of the slab in the shallow upper mantle appears to be
firstly controlled by the strength of the surface plate and is not sensitive to the
weakening of the slab. This is best demonstrated in
Figs.

The effects of a lower mantle on subduction and the extent to which the lower
mantle participates in convection are debated on both geophysical and
geochemical grounds

1550 K isotherm for models 3 (left/orange), 5 (center/green), and 7
(right/blue). Surface plates are 50 km thick and have a maximum viscosity of

The concept of steady-state subduction as invoked in scientific literature is
variously defined depending on the context of a particular paper. Articles
that examine earthquake cycles consider the long-term motion of the plates to
be steady-state as compared to the stick and slip of earthquakes that take
place on timescales of decades or centuries

Our evaluation of three methods for calculating radius of curvature showed that its time-varying nature is related to the state of the subducted slab. The spline method is subject to variability in models due to its nature and selection of a smoothing parameter. The spline method results in a smaller radius of curvature than either of the two circle methods. The circle methods are sensitive to changes of the morphology of the subducted slab, while the spline method is not. The method of fitting a circle to slab shape from the surface to a depth of 150 km is sensitive to the state of the subducted slab below 150 km, covers a wide range of radii of curvature, and is less noisy than other methods.

The strength of plates and slabs independently controls the shape of the
shallow slab and subducted slab. The richness of slab morphology as seen in
tomographic inversions suggests that the mechanical strength of tectonic
plates is modified by some mechanism. As in

Strong plates resist bending and preserve larger radii of curvature. Smaller radii of curvature, the result of vertical descending or steeply dipping slabs, promote two-sided convection. Larger radii of curvature preserve single-sided subduction. At depth, strong slabs embed in the lower mantle, transmit bending stresses to the surface, and decrease radii of curvature. Weak slabs at depth are less coupled to the lower mantle, and inhibit the transmission of stress to the surface.

Earth-like subduction is single-sided and supports a wide range of radii of curvature and slab morphologies. In this study, the richness of such observations is best reproduced by a combination of strong surface plates and weakened slabs. The strong plates promote single-sided subduction while the weakened slabs allow slabs to bend, pile, or lie flat, as is seen in tomographic inversions.

We are grateful to Boris Kaus and an anonymous reviewer for their constructive comments. This material is based upon work supported by the National Science Foundation under grant no. 1255040. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation grant number ACI-1053575. The authors acknowledge the Texas Advanced Computing Center (TACC) at the University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. Edited by: J. C. Afonso Reviewed by: B. J. P. Kaus and one anonymous referee