The influence of plate boundary motion on planform in viscously stratified mantle convection models

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2011jb008362, 2011 The influence of plate boundary motion on planform in viscously stratified mantle convection models J. P. Lowman, 1,2 S. D. King, 3 and S. J. Trim 4 Received 10 March 2011; revised 20 September 2011; accepted 23 September 2011; published 8 December [1] A number of studies examining the influence of plates on mantle convection have concluded that planform and thermal structure are strongly influenced by plate geometry. However, studies that have modeled evolving plate geometries over periods greater than a mantle transit time indicate that mantle planform may not correlate with plate geometry. To assess the influence of plate boundary motion on mantle convection, we investigate convection in a plane-layer system featuring four polygon-shaped plates. New to this work, plate boundaries are moved at specified velocities that are consistent with the velocities associated with the convection driven flow in the system interior. Plate velocities are time dependent and use a force-balance method to ensure that plate motion neither drives nor resists the convection. The influence of the plate boundary motion on convection is compared in models featuring viscosity profiles that increase by factors of 30, 90, 300, and 1000 across the lower mantle. The effective Rayleigh numbers of these systems are held at a nearly constant value. We find that convection planform is sensitive to both divergent and convergent plate boundary motion for a system featuring as much as a 90-fold contrast in viscosity between the upper and lower mantle. However, as the viscosity stratification is increased, the response of the convection planform to the motion of divergent plate boundaries diminishes. In contrast, we find planform and specifically plume positions respond to the motion of convergent plate boundaries even when the lower mantle viscosity is 1000 times greater than the upper mantle viscosity. Citation: Lowman, J. P., S. D. King, and S. J. Trim (2011), The influence of plate boundary motion on planform in viscously stratified mantle convection models, J. Geophys. Res., 116,, doi: /2011jb Introduction [2] While the pioneering work of Oxburgh and Turcotte [1968] has shown that mantle convection has sufficient energy to drive tectonic plates at their observed velocities, a complete physical theory that predicts and explains the full range of observations that make up our understanding of plate tectonics remains elusive. Forsyth and Uyeda [1975] showed that plate velocities strongly correlate with the percentage of plate circumference corresponding to a subduction zone (i.e., slab pull) and to a lesser extent the percentage associated with ridges (i.e., ridge push). More recently, Schellart et al. [2010] found subducting-plate velocity increases with subduction zone width. Such observations, coupled with the very weak correlation between plate area 1 Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada. 2 Also at Department of Physics, University of Toronto, Toronto, Ontario, Canada. 3 Department of Geosciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA. 4 Department of Physics, University of Toronto, Toronto, Ontario, Canada. Copyright 2011 by the American Geophysical Union /11/2011JB and plate velocity, have lead to and subsequently supported thinking of plate boundary forces as the driving component of plate motions. Plate motions can be reproduced by viscous flow driven by buoyancy from subducted slabs [Hager and O Connell, 1981] or density anomalies derived from global seismic tomography models [e.g., Forte and Peltier, 1987; Conrad and Lithgow-Bertelloni, 2002]. In such models, plates are often viewed as passive rafts riding atop a convecting mantle because the tractions from the mantle flow beneath the plate drive the plate motion. However, it is clear that the end-member views of plates as passive riders atop a convecting mantle or an independent system only loosely connected to mantle convection are insufficient for a complete physical theory of plate tectonics. While it has become widely accepted that the plates are an integral part of mantle convection [e.g., Davies and Richards, 1992; Bercovici et al., 2000], whether the location of plate boundaries is controlled by processes deep within the Earth (i.e., length scales imposed by the convecting cells) or is the result of mechanical properties (i.e., faults or weak materials), remains an open question [Anderson, 2001]. [3] There are several approaches to plate modeling in mantle dynamics. One approach uses a viscoplastic constitutive theory to produce plate-like behavior [e.g., Bercovici, 1993; Trompert and Hansen, 1998; Tackley, 1998, 2000a; 1of14

2 Stein et al., 2004; Stein and Hansen, 2008; van Heck and Tackley, 2008], while the other approach imposes a priori faults [Zhong and Gurnis, 1995], weak zones [e.g., Gurnis and Hager, 1988; King and Hager, 1994; Chen and King, 1998; Foley and Becker, 2009; Stadler et al., 2010], or plate geometries [Gable et al., 1991; Monnereau and Quéré, 2001; Lowman et al., 2001, 2003, 2004; Conrad and Lithgow- Bertelloni, 2002; Quéré and Forte, 2006; Brandenburg and van Keken, 2007; Ghias and Jarvis, 2007]. Each of these approaches has advantages and disadvantages. At present, constitutive theory based calculations of convection with mobile plates have yet to produce plate-like behavior over many mantle overturns, limiting their application to problems where long-term evolution of the flow are not important. The a priori imposed plate boundary methods can produce self-consistent plate-like motions for many mantle overturns; however self-consistent plate boundary evolution remains a challenge for these methods [Gait and Lowman, 2007a, 2007b; Lowman et al., 2008; Gait et al., 2008; Stein and Lowman, 2010]. [4] Changes in plate velocity (both speed and direction) have been described as plate reorganization events. Previous work [King et al., 2002; Ghias and Jarvis, 2007] has shown that the velocity of mobile plates can change (speed and direction) over a period of less than 5 Myr in response to changes in the fluid system, consistent with the spatial and temporal characteristics of plate reorganizations on Earth. Furthermore, the location of upwelling plumes can remain unaffected by plate reorganizations [Lowman et al., 2004]. One limitation of the previous work is that while the plate direction and speed evolved consistent with the underlying fluid flow, the location of the plate boundaries in these calculations remained fixed throughout the calculation. One could speculate that if the plate boundaries in those calculations evolved in a self-consistent manner in response to changes in the underlying planform of the convecting fluid, it might reduce or eliminate the need for the observed dramatic changes in plate velocity. [5] In this study, we describe a series of Cartesian geometry convection calculations featuring both fixed and mobile plate boundaries. Unlike our previous work that had either fixed [Lowman et al., 2001; King et al., 2002; Lowman et al., 2003, 2004] or self-consistent plate boundary evolution [Gait and Lowman, 2007a; Lowman et al., 2008; Gait et al., 2008; Stein and Lowman, 2010], here we impose a fixed rate of plate boundary motion to a series of models with dynamically controlled plate motion in an attempt to isolate the effects of dynamic plate velocities and plate boundary motion. We analyze 12 convection models featuring different ratios of lower to upper mantle viscosity and compare six pairs of calculations (with and without plate boundary motion), where each pair of calculations begins from the same initial conditions. Regardless of whether mobile plate boundaries are specified, in all experiments the plate velocities evolve dynamically in a self-consistent way that neither imparts energy to, nor retards the motion of the plate-mantle system [e.g., Gable et al., 1991; Lowman et al., 2003]. [6] To assess whether a lower mantle viscosity that is high relative to the upper mantle viscosity is able to exhibit a thermal inertia that decouples from plate evolution (effectively retaining a memory of earlier plate configurations that cannot keep pace with changing surface conditions) we vary the contrast between the upper mantle and lower mantle viscosities while holding the effective Rayleigh number approximately fixed. (Our use of the term effective Rayleigh number refers to the collective parameters used in the Rayleigh number and internal heating rate. For a fixed nondimensional heating rate, in systems with different viscosity stratifications we consider the effective Rayleigh number to be unchanged when the same mean heat flux is obtained. Our methodology for obtaining equivalent effective Rayleigh numbers is explained in section 2.) We compare changes in the deep mantle planform in calculations with fixed and mobile plate boundaries and identify cases where existing planforms conflict with the evolving plate geometry. 2. Model Description [7] In order to simulate mantle convection we model an infinite Prandtl number, Boussinesq fluid with a stratified Newtonian rheology. We employ a Cartesian geometry solution domain. The nondimensionalized equations governing the evolution of the convection are derived from the conservation of mass, momentum, and energy and take the forms v ¼ 0; ðhðzþ Þ P¼ Ra B Tz^; T t ¼ 2 T v T þ Hz ðþ; respectively. The quantities in equations (1) (3) are the velocity v, the depth-dependent dynamic viscosity h(z), the strain rate tensor, pressure P, temperature T, and time t. Ra B is the Rayleigh number [Chandrasekhar, 1961] and is given by Ra B ¼ rgadtd3 kh um where r is density; h um is the dynamic viscosity of the upper mantle; g is gravitational acceleration; a is thermal expansivity; DT, is the superadiabatic temperature difference between the top and bottom boundaries; d, is the depth of the convecting layer and k, is the thermal diffusivity. [8] We model geochemically homogeneous convection by implementing a depth-dependent internal heating rate, H(z). The layered heating simulates the reduction in volume of a finite thickness spherical shell as a function of radius. We compare a spherical shell segment and a Cartesian layer with the same thickness and equate the total heat production rate and the heat production rate at the surface in order to obtain a depth-dependent heat source distribution in the plane-layer geometry case [Nettelfield and Lowman, 2007]. The nondimensional heating rate at the top of the calculations is 15. The mean heating rate in the models is The dimensional values used to estimate that H = 15 were previously given by Lowman et al. [2001]. ð1þ ð2þ ð3þ ð4þ 2of14

3 Figure 1. Plate geometry snapshots from the models featuring evolving plate boundaries. The labels marked in degrees indicate the angles through which the mobile plate boundaries have rotated (counterclockwise) relative to their starting position (i.e., 0 ). The axis of rotation is normal to the plane of the figure and passes through the center of each panel. Plate boundaries around the perimeter of each panel remain stationary. The color of each panel corresponds to the color contours used in Figures 3, 6, 8, and 9. [9] The vigor of the dynamically driven flow is determined by the specified Rayleigh number and internal heating rates, as well as the depth-dependent viscosity profile. The lower mantle viscosity in all calculations increases logarithmically (details are given by Gait and Lowman [2007a]) from its nondimensionalized upper mantle value of 1.0 to a value of 30, 90, 300 or 1000 at the base of the mantle. [10] The Rayleigh numbers we implement in calculations with viscosity contrasts of 30, 90, 300, and 1000 are , ,2 10 8, and , respectively, based on the upper mantle viscosity in the calculations. In the final two cases (i.e., viscosity contrasts of 300 and 1000) we prescribe the same Rayleigh number because of the constraints of our computational resources and the grid resolution required to model such a weak upper mantle. On the assumption that, other than the reference upper mantle viscosity, all Rayleigh number parameters are held constant, changes in the Rayleigh number are inversely proportional to changes in the upper mantle viscosity. Consequently, comparing a reference model with an increase in viscosity across the lower mantle of a factor of 30, with models where viscosity increases by a factor of 90, 300 or 1000 (with Rayleigh numbers, using the definition in equation (4), of ,1 10 8, and , respectively) implies that the upper mantle in the three latter cases are one half, one quarter, and one quarter, respectively, of the upper mantle viscosity of the reference model. Accordingly, the lower mantles are 1.5, 2.5, and 8.3 times more viscous. [11] Connected (i.e., wrap around) flow is modeled by implementing a periodic solution domain. A free-slip basal boundary condition is specified and the convecting layer is confined between isothermal horizontal boundaries in all calculations. Calculations are performed in 3-D rectangular geometries with dimensions of using computational grids that have nodes. The middepth area of our Cartesian calculations simulates half of the area of a sphere with a radius equal to core plus midmantle depth. MC3D, a hybrid spectral finite difference scheme for 3-D convection previously described by Gable et al. [1991] has been used to solve the system of equations (1) (3). MC3D has been benchmarked for a variety of plane-layer convection problems and in those cases shows excellent agreement with the results obtained from other numerical methods [e.g., Travis et al., 1991; Busse et al., 1993]. [12] We implement a force balance plate modeling method to achieve plate-like surface motion [Gable et al., 1991]. Distinct plates are defined by specifying regions of spatially uniform velocity at the surface of the solution domain. Selfconsistent plate motion is determined by calculating stress on a plane defining the base of the plate. Stresses resulting from buoyancy-driven flow both below the plate and within the plate are calculated with a no-slip surface boundary condition. These stresses are balanced against shear stresses from purely plate-driven flow. The plate velocity is determined by the condition that the integrated shear stress due to the plate motion must equal the integral of the buoyancy induced driving shear stress. The superimposed buoyancydriven and plate-driven flows result in plate motion that is in dynamic equilibrium with the buoyancy-driven flow so that the only force acting on the plate is the integrated buoyancy within the plate and mantle. Forces associated with slab pull and ridge push are incorporated into our continuum calculations implicitly through the buoyancy associated with the downwelling cold limb of the convecting fluid and the thickening of the upper thermal boundary layer, both of which contribute to the balance of forces that drive our plates [cf. Davies and Richards, 1992; Bercovici et al., 2000; King, 2001, 2007]. [13] The temporally and spatially constant-thickness plates are 1000 times more viscous than the underlying upper mantle and compose 4.0% of the total system depth, or approximately 120 km. This high-viscosity layer is coincident with the upper thermal boundary layer, which is of similar mean thickness in nearly all calculations because of our maintaining a roughly constant effective Rayleigh number throughout the study. While our viscosity is not temperature dependent, the high-viscosity layer acts as a first-order approximation for the stiffness associated with the Earth s cold lithosphere and leads to an asymmetry in the temperature field with larger temperature contrasts between the downwellings and ambient mantle than the upwellings and ambient mantle. The downwelling slabs are the same strength as the surrounding mantle. As shown by King and Ita [1995] weak slabs are more easily deformed than strong slabs by an increase in viscosity or a phase change. The calculated plate velocities evolve to reflect the changing distribution of buoyancy within both the plates and underlying fluid. The force balance modeling method has been shown to reproduce very similar velocities and surface heat fluxes to methods that model plates by implementing non-newtonian viscosities or specified stiff lithospheres with weak plate boundaries [e.g., King et al., 1992; Koglin et al., 2005]. [14] We examine the effect of plate boundary motion on mantle flow by comparing calculations featuring stationary and mobile plate boundaries. The motion of the plate boundaries in the later case is prescribed, rather than calculated on the basis of the dynamics of the systems [e.g., Gait et al., 2008; Stein and Lowman, 2010], however, in all cases the plate velocities are dynamically determined by the force balance criterion described above. [15] Figure 1 illustrates the positions of the evolving plate boundaries at the start of the calculations and after 3of14

4 each 15 increment in the plate boundary rotation. (The different colors used for the plots are relevant to Figures 3, 6, 8, and 9 and are explained below.) Experiments characterized by fixed plate boundaries maintain the plate boundary positions shown in the 0 plot in Figure 1. In the calculations featuring mobile plate boundaries, the boundaries steadily rotate in the surface plane around the point at x =2,y =2,z = 1. However, the two plate boundaries specified at x = 0 and y = 0 do not move. To simplify the calculations, the plate boundaries move with discrete rotations of 1 and maintain their position for a time Dt following each degree of rotation (where Dt is roughly the time required to travel d at the mean plate velocity). After each period Dt the boundaries shift to complete another 1 of rotation and maintain their new position for a subsequent Dt. A full 90 of rotation is completed in each calculation. Assuming a thermal diffusivity of 10 6 m 2 /s and mantle depth of 2900 km implies Dt = 6.66 Myr. [16] In order to obtain an initial thermal field for each of the viscosity contrasts investigated, we first solve for twodimensional (x,z) convection employing the specified viscosity models, internal heating rate and Rayleigh numbers described above. The 2-D solutions are obtained in an aspect ratio 4 model featuring a pair of equal size plates and are integrated forward in time until reaching a point where they exhibit neither long-term heating nor cooling trends. The two-dimensional solutions are then projected in the third (y) dimension to produce fields that occupy volumes. The 3-D fields are subsequently integrated forward in time with the static plate boundaries shown in the 0 plot in Figure 1. [17] By specifying different initial thermal fields in the two-dimensional start-up calculations we can obtain downwellings in different locations. These downwellings subsequently produce convergent boundaries in different locations when we extend the two-dimensional fields to produce three-dimensional initial conditions. Accordingly we are able to obtain initial 3-D fields with convergent boundaries in the desired locations. [18] In section 3 we divide our study into cases featuring predominantly convergent or divergent mobile plate boundaries. The setup of the initial condition fields is thus chosen in order to obtain specific initial planforms. Experiments start from arbitrary temperature field snapshots. However, initial temperature fields for the 3-D experiments are not chosen until the time-dependent trial solutions have again reached a state where any long-term heating or cooling has ceased. 3. Results 3.1. Initial Conditions and Plate Boundary Motion Determination [19] We first consider models with convergent plate boundaries at y = 0 (or, given the periodicity of the calculations, y = 4). The same initial conditions are used for each pair of models with fixed plate boundaries and evolving plate boundaries. From this point we shall refer to these calculations as models AX, where X = 300, 90, or 30 and indicates the factor by which the lower mantle viscosity increases from top to bottom. We shall also add the suffix f or e to a model name to denote whether a calculation features fixed or evolving boundaries. [20] The initial temperature fields obtained for model A300(f,e), A90(f,e) and A30(f,e) exhibit similar planforms, characterized by upwellings (clustered along a vertical plane corresponding to y = 2) and sheet-like downwellings plunging into the mantle along the plate boundaries coinciding with the y = 0 plane. In addition, downwellings are present at the plate boundary separating plates 1 and 2 (as defined in Figure 1, 0 plot). [21] Models featuring initially divergent boundaries at x = 0 and y = 0 shall be referred to as models B90(f,e), B300(f,e) and B1000(f,e). The mobile plate boundaries in models B90e, B300e and B1000e are associated with convergent plate boundaries. [22] We investigate how mantle upwellings respond to divergent and convergent plate boundary migration for different mantle viscosity structures by examining pairs of calculations with and without specified plate boundary motion. For each viscosity profile we first analyzed the evolution of the calculation with the fixed plate geometry to determine the mean plate velocity. Over the duration of the fixed boundary calculations presented, the mean plate velocities of all of the models differ by less than 10%, suggesting that we have obtained similar convective vigor for each viscosity model considered. We choose a rate of plate boundary motion on the basis of the mean of the plate velocities from the fixed plate geometry models. The plate boundary migration rate is the same in models A30e, A90e, B90e, A300e, and B300e. The plate boundaries move at half this rate in model B1000e, in which the effective Rayleigh number is lower. [23] The fastest moving plate boundary sections are at the plate boundary triple junctions that migrate along the boundaries fixed at x = 0 (and 4) and y = 0 (and 4). In models A30e, (A and B)90e, and (A and B)300e, these sections of plate boundary move with a mean velocity approximately equal to the mean plate velocities in models A30f, B90f and B300f. In each calculation, the total distance traveled by the triple junction is equal to four times the system depth, or four surface transit times. [24] Measured in nondimensional time, models A30(e and f), A and B90(e and f) and A and B300(e and f) simulate a period of diffusion times (or 600 Myr on the basis of a thermal diffusivity of 10 6 m 2 /s and a mantle depth of 2900 km). Accordingly, the fastest moving mobile plate boundaries in our calculations average a speed of 2 cm/yr and the mean plate velocities are also roughly 2 cm/yr, somewhat low relative to the global average for the Earth because of the lower effective Rayleigh number of the calculations Experiments Mobile Divergent Plate Boundaries [25] In Figure 2 we show snapshots of the temperature fields from three calculations. The top 5% of the temperature fields have been removed to allow a view of the model interiors. Figure 2 (left) shows temperature fields from model A300f, and Figure 2 (middle) shows temperature fields from model A300e. In order to determine whether plate boundary movement has an effect on mantle planform after some lag, we have performed a third experiment in which the final plate geometry from the evolving plate 4of14

5 Figure 2. Snapshots of temperature isosurfaces from three experiments: (left) model A300f, (middle) model A300e, and (right) model A300c. Hot (green) and cold (blue) isosurfaces correspond to temperatures of 0.68 and Also shown are the plate geometries and relative plate velocity direction and magnitudes at the times corresponding to each temperature field snapshot. The length of the velocity vectors is proportional to the velocity vector shown, which has a length corresponding to 5000 transits of the system depth per diffusion time (about 5.42 cm/yr). A set of axes indicates the x and y directions referred to in the text. The range of x and y coordinates covered is 0 4. The first row shows the initial condition for each model. Each row corresponds to the same time relative to the start of the experiment. For example, the panel in the fourth row of Figure 2 (middle) shows the temperature field from model A300e after the mobile plate boundaries have rotated through 90. The panel in the fourth row of Figure 2 (left) shows the temperature field from model A300f after the exact same period but in the case where the plate boundaries have all remained fixed. Given the conversion to dimensional values described in the text, dt corresponds to a period of 200 Myr. geometry case is held fixed and the final temperature field from the evolving boundary calculation is specified as the initial condition. We shall refer to these calculations as models A30c, (A and B)90c, (A and B)300c and B1000c. Figure 2 (right) shows the evolution of model A300c. [26] Figure 2 includes plots of the plate geometries corresponding to each temperature field snapshot. The first row of Figure 2 shows the initial conditions for each model. Superimposed on the plate geometry maps are arrows indicating the direction of the plate velocity. Arrow length indicates the magnitude of the plate velocity according to a linear scale calibrated to the length of the arrow at the top of Figure 2. (The scale arrow should be interpreted as meaning that a plate moving at a velocity represented by this length will traverse a distance equal to the system depth, d, 5000 times per diffusion time, corresponding to roughly 5.42 cm/yr assuming a thermal diffusivity of 10 6 m 2 /s.) The initial temperature field for Figure 2 (right) is the bottom temperature field in Figure 2 (middle). The three temperature field snapshots shown in Figure 2 (middle) correspond to the times at which the mobile plate boundaries have rotated by 30, 60, and 90 (rows labeled t = dt, 2dt, and 3dt, respectively) about the center point of the plate geometry map. The snapshots in Figure 2 (left) correspond to the same times, but from model A300f. The snapshots in Figure 2 (right) for 5of14

6 Figure 3. Contours of vertically advected heat flux at seven different times in models (left) A300f, (middle) A300e, and (right) A300c. The plotted color contours are from times corresponding to the moment at which the moving plate boundaries in model A300e take on the geometry shown by the panels in Figure 1 of the same color. (top) The UM row shows the contour of nondimensional value 100 in the upper mantle at 12% of the system depth (i.e., middle of upper mantle). (bottom) The LM row shows the same value contour in the lower mantle at two thirds of the full system depth. Figure 4. Initial and final fields from model A90f, model A90e, and model 90c. Isosurface values are 0.73 and of14

7 model A300c have the same temporal separation as in Figures 2 (left) and 2 (middle). [27] Figure 2 shows that although model A300f is time dependent, the (convection pattern) planform and plate velocity directions change very little during the period modeled. More surprisingly, the planform changes very little during the evolution of model A300e and in the period that follows the plate boundary movement (model A300c), the convection planform change is also minimal (i.e., upwellings remain clustered along the vertical plane y = 2 and the dominant downwelling is at y = 0). [28] The upwellings in models A300(f, e and c) have a structure that features broad conduits with several adjoining sheet-like upwellings extending to approximately middle lower mantle height. We find that more columnar, plumelike, upwellings are obtained as the contrast between upper and lower mantle viscosity is reduced. [29] To track upwelling motion we calculate the vertically advected heat flux (VAHF). The VAHF is the component of the vertical heat flux associated with advection as opposed to diffusion and is only positive when velocities are positive. Thus high values identify hot rising material explicitly. For the models with evolving boundaries, VAHF contours are used to identify plume locations at the times corresponding to the arrival of the plate boundaries at the locations shown by the plate geometry maps in Figure 1. VAHF is also calculated for the corresponding models in which the plate boundaries are held fixed. VAHF contours are also plotted for the calculations that follow the period when plate boundary evolution ceases ( c models). [30] Figure 3 shows contours of the VAHF in models A300f (Figure 3, left), A300e (Figure 3, middle), and A300c (Figure 3, right). The contours plotted have a nondimensional value of 100; this value outlines plume locations. Figures 3 (top) and 3 (bottom) show contours on horizontal planes at approximately 12% and 67% of the total system depth, respectively. The former depth corresponds to the depth of the middle upper mantle. The colors of the contours in Figure 3 correspond to the colors of the maps in Figure 1. For example, the green contours are the contours corresponding to a VAHF of 100 at the time when the mobile plate boundaries in model A30e have moved to the positions labeled as 45 in Figure 1. [31] If plume locations remain fixed during an experiment, then the contours in all colors will lie in the same location. Thus the different colors will produce the same pattern. If the plumes move then the different contour colors will produce either a trail or random pattern, depending on the overall time dependence of the flow. [32] Figure 3 (left) shows that model A300f is characterized by a very time dependent flow (indicated by the distinct shape of the contour corresponding to each time) but an essentially fixed pattern, where plumes generally occupy the same limited range of locations. Several isolated contours appear but they are not common and are generally associated with transient developing instabilities that get drawn into clusters of plumes. The VAHF contours show the plume locations in model A300c remain in similar locations to the plumes in model A300f, indicating that the change in plate boundary position has not had a significant impact on upwelling locations. [33] In Figure 4 we show snapshots of the initial and final temperature fields from models A90f, A90e, and A90c along with corresponding plate geometry maps and plate velocities. Each experiment is of the same duration as the cases depicted in Figure 2. As in model A300f, the pattern of convection in model A90f does not change, although the convection itself is highly time dependent. The final convection pattern in model A90e is also unchanged from the initial pattern (i.e., the planform is characterized by a downwelling sheet at y = 0 and upwelling plumes gathered along a vertical plane at y = 2). However, the panel for the last time step from model A90c in Figure 4 shows there is a delayed response of the flow to the motion of the plate boundaries and imposition of a new plate geometry. Flow reorganization is driven by a change in direction of the plate velocities that results from a mismatch of the plate geometry and underlying convection pattern. As the boundaries move, the influence of the plate motion on the deep convection pattern changes and the original convection pattern is no longer consistent with surface motion. Moreover, as plate velocities change, plate boundaries can change from divergent to convergent and vice versa. Once a new convergent boundary forms its development rapidly increases as downwelling (slab-like) features emerge and pull the plates in the direction of the boundary. Consequently, surface velocities and the interior convection pattern can reorganize quickly in response to the moving plate boundaries. [34] In Figure 5 we show snapshots of the temperature fields and corresponding surface velocities from models A30f, A30e, and A30c. Although model A30f is time dependent, the planform and plate velocity directions change very little during the period modeled. In contrast, the effect of the plate boundary motion in model A30e is dramatic. The overall planform of the convection differs very little in the models after 30 and even 60 degrees of plate boundary migration (in fact, many of the plumes in model A30f are clearly observable in model A30e, at times t = dt and 2dt) however, by the end of the calculation the system has responded to the plate boundary evolution by undergoing a complete flow reorganization. This reorganization occurs even though the position of the dominant downwelling in the calculation (along x = 0) is not forced to move by the plate geometry evolution. Once the plate boundary motion ceases (Figure 5, right) a new steady convection pattern is established featuring time-dependent flow. Plate reorganization is thus driven by the movement of the plate boundaries while steady planforms are obtained with immobile plate boundaries. For upper mantle viscosities that are 90 and 30 times less than the viscosity in the deep mantle, divergent plate boundary motion is capable of causing a reorganization of deep mantle convection. However, for an upper mantle viscosity that is 300 times lower than the deep mantle viscosity we find that plate boundary motion and accompanying changes in plate velocity have little influence on the pattern of convection in the deep mantle. [35] The VAHF contours plotted in Figure 6 confirm that the plume locations in model A30f (Figure 6, left) are highly time dependent but that the plumes repeatedly appear in roughly the same locations. Figure 6 (middle) (corresponding to model A30e) is similar to the plots from model A30f (particularly in Figure 6, bottom), indicating that plume 7of14

8 Figure 5. As in Figure 2, but depicting the evolution of (left) model A30f, (middle) model A30e, and (right) model A30c. Isosurface values are 0.80 and Given the conversion to dimensional values described in the text, dt corresponds to a period of 200 Myr. locations were not changed substantially by the plate boundary motion in model A30e. However, the blue and black contours from model A30e are quite different from model A30f, indicating a difference in the evolution of the experiments in their latter stages. The contours from model A30c take on a completely different arrangement from those in model A30f, indicating that the plumes have relocated because of the plate reorganization event that occurred during the evolution of model A30e. Moreover, the scattered distribution of the contours in the panel corresponding to model A30c indicates that a stable planform was not established during the time model A30c evolved. Figure 6 (top) shows similar evolution to Figure 6 (bottom). The weakness of the plumes at this depth does not allow the VAHF contours to delineate such a strong pattern as the contours from the lower mantle, but the change in the contour distributions in Figure 6 (top) clearly shows that the plate evolution episode caused the upper mantle plumes to relocate Mobile Convergent Plate Boundaries [36] We now turn attention to experiments featuring initially convergent mobile plate boundaries. The first row of Figure 7 shows initial temperature field snapshots from nine calculations featuring viscosity contrasts of 90 (Figure 7, left), 300 (Figure 7, middle), and 1000 (Figure 7, right). The second, third, and fourth rows show the final fields from calculations featuring fixed plate geometries, evolving plate boundaries and cases started from the final plate geometry of the evolving plate boundary models, respectively. Also shown are the corresponding plate geometry maps with arrows indicating the direction and magnitudes of the instantaneous plate velocities. [37] In contrast to the models previously described, models B90f and B300f have somewhat time-dependent planforms. Nevertheless, in general the upwellings in the models remain confined to limited regions and the boundary at y =0 remains divergent. However, when mobility of the boundaries in these models is enabled dramatic changes in planform are initiated [e.g., Gait et al., 2008; Stein and Lowman, 2010]. Even with a factor of 300 viscosity contrast across the lower mantle, the movement of the sheet-like downwellings stirs the lower mantle and the upwelling features observed in the initial condition (Figure 7, first row) slowly move to new positions. At the end of the sequence of plate boundary motion, completely new planforms are established in both models. However, unlike in model A30e or B90e, plate velocity time series show that model B300e does not 8of14

9 Figure 6. As in Figure 3, but for models (left) A30f, (middle) A30e, and (right) A30c. respond to the plate boundary motion with a flow reorganization event characterized by plate reversals or any plate velocities temporarily dropping to zero. Subsequent to the period of plate boundary motion a steady change in direction of the plates occurs, resulting in a completely new convection pattern (fourth row in Figure 7, middle) in comparison to model B300f. [38] Model B1000f exhibits a very steady convection pattern throughout. However, despite the extremely high viscosity contrast between the upper and lower mantle, the final temperature field of model B1000e shows a response to the plate motion. The response of the plumes to the relocation of the downwellings is gradual but clear at the end of the final calculation (bottom, right). In all cases examined, mobile plate boundaries associated with downwelling flow cause plume locations to gradually move. [39] The foregoing suggests that the effect of plate boundary motion on deep mantle convection patterns (and therefore plume motion) is determined by both the viscosity contrast [Steinberger and O Connell, 1998] between the upper and lower mantle and the type of plate boundary in motion, divergent or convergent. The motion of convergent plate boundaries on deep mantle flow is evident even when the viscosity contrast of the upper and lower mantle reaches a factor of [40] Figure 8 shows VAHF contours from models B300f, e, and c. Figure 8 (bottom left) shows that in model B300f there are two main plume complexes positioned in close proximity to each other (these are centered at approximately x = 0.25, y = 0.25 and x =0,y = 3.25). In model B300f upwelling flow either starts at or is drawn into these two locations. In model B300e, the impact of the motion of the downwellings is shown by the migration of the contours in Figure 8 (bottom middle). The contours associated with the plumes can be seen to slowly migrate toward a vertical plane along y = 2. The contours trace steady and gradual plume repositioning rather than the disappearance of the initial plumes and emergence of different plumes. In model B300c (Figure 8, right) the plume motion slowly continues as the lower mantle adjusts to the change in plate geometry. In general, compared with the contours from the lower mantle, the plume conduits in the upper mantle show less coherent continuous plume motion in both models B300e and B300c. [41] The VAHF in models B1000f, B1000e, and B1000c are shown in Figure 9. Just two plume complexes exist in model B1000f. The plume locations are steady in model B1000f (Figure 9, left), gradually migrate in model B1000e (Figure 9, middle), and coalesce to produce a new planform in model B1000c (Figure 9, right). In contrast to models B300e and B300c, the motion of the plumes is easier to follow in both the lower and upper mantles. 4. Discussion [42] In the calculations presented, the motion of the plate boundaries results from an externally applied force, allowing us to create a controlled set of experiments where we can 9of14

10 Figure 7. Initial condition and final snapshots of temperature isosurfaces from nine experiments. Hot and cold isosurface pairs are (0.71, 0.47), (0.69, 0.45), and (0.62, 0.38) in the panels depicting models B90(f, e, and c), B300(f, e, and c), and B1000(f, e, and c), respectively. Results from models (left) B90(f, e, and c), (middle) B300(f, e, and c), and (right) B1000(f, e, and c). assess the impact of the motion of plate boundaries on the planform of convection in the lower mantle. Plate boundary mobility introduces a key aspect of terrestrial mantle convection omitted in plate calculations with fixed plate geometries [e.g., Monnereau and Quéré, 2001; Lowman et al., 2001, 2003, 2004] and a dramatic effect on system evolution. The results presented here indicate that mantle plume positions, [e.g., Tarduno et al., 2003; Steinberger et al., 2004; Tarduno, 2007] plate velocity direction and magnitude are all affected by plate boundary motion. [43] One of the more interesting results from these calculations is the difference in the response of the lower mantle to evolving convergent and divergent plate boundaries. This difference is consistent with the conventional view in plate tectonics that ridges are passive features [e.g., Bercovici et al., 2000]. In our calculations, with the exception of the smallest upper mantle/lower mantle viscosity contrasts, the lower mantle convective planform is almost unaffected by evolving divergent boundaries. This is consistent with the observation that certain hot spots, such as the Galapagos, presumed to be the surface manifestations of plumes originating from the lower mantle, can jump across ridge boundaries [Sinton et al., 1996] and with seismic tomographic models that suggest anomalies beneath most midocean ridges are confined to the upper mantle, possibly extending into the transition zone [Zhou et al., 2006]. [44] In contrast, we find converging plate boundaries have a significant effect on the lower mantle convective planform, even with the largest upper mantle/lower mantle viscosity contrasts considered. If we adopt the conventional view that fast seismic velocity anomalies represent cold, downwelling convection limbs, this is consistent with the observation that fast seismic velocity anomalies that reach the surface at trenches can be readily identified in seismic tomograms extending into the transition zone and beyond [e.g., van der Hilst et al., 1997; Miller and Kennett, 2006] and fast 10 of 14

11 Figure 8. As in Figure 3, but for models (left) B300f, (middle) B300e, and (right) B300c. anomalies in the lower mantle that can be linked to the location of subduction zones in the past [e.g., Van der Voo et al., 1999]. [45] Although generally the viscosity contrasts used in these calculations are within the range constrained by geophysical observations [e.g., King and Masters, 1992; King, 1995; Mitrovica and Forte, 2004], we include one case with a 1000-fold viscosity contrast between the asthenosphere and lower mantle, to provide an end-member case with a relatively sluggish lower mantle that is resistant to the changing upper mantle thermal structure. Changing the viscosity contrast in these calculations decreases the viscosity of the asthenosphere relative to the deep mantle. Decreasing the ratio of upper to lower mantle viscosity increases the convective vigor of the upper mantle relative to the lower mantle. As the viscosity contrast increases, the time required for the upper mantle to adapt to the evolving plate geometry decreases, while the planform of convection in the lower mantle takes longer to evolve to a pattern that is consistent with the new plate geometry. The upper mantle is caught between the evolving plate geometry above and the slowly responding lower mantle below [e.g., Anderson, 2001]. [46] It is worth remembering that we do not have temperature-dependent viscosity, so our slabs are weak. We anticipate that stronger slabs would have a stronger effect on lower mantle planform because the slabs would be more resistant to deformation from the surrounding mantle. An additional feature not considered in our models is the presence of a low-viscosity boundary layer at the base of the mantle. Such a layer could correspond to a thermal boundary layer [e.g., Stacey and Loper, 1983; Hager and Richards, 1989; Lay et al., 1998; Lassak et al., 2007] or the transition to postperovskite [e.g., Takashi and Yamazaki, 2007; Cizkova et al., 2009; Ammann et al., 2010; Nakagawa and Tackley, 2011] and could possibly allow greater plume mobility. However, in the Earth the characteristics of the region enveloping the core (specifically D ) may be very heterogeneous [e.g., Lay et al., 1998; Thomas et al., 2004; Hernlund et al., 2005; Lay et al., 2006; van der Hilst et al., 2007; Wenk et al., 2011; Nakagawa and Tackley, 2011] because of the possible presence of thermochemical piles beneath Africa and the Pacific [e.g., Trampert et al., 2004; Ni et al., 2002; To et al., 2005; Lassak et al., 2007, 2010], and/or a slab graveyard [e.g., Maruyama et al., 2007; Spasojevic et al., 2010]. Given the uncertainties surrounding the complexities in this region, we simply maintain the viscosity of the overlying mantle through D down to the base of the system. Still, we note that the presence of a lowviscosity layer at the base of the mantle could have implications for the location and dynamics of plumes [e.g., Tosi et al., 2010]. Increased upwelling mobility should strengthen our conclusion that upwelling positions will respond to the motion of convergent plate boundaries. However, it could also allow for the possibility that divergent plate boundary motion might affect plume motion for stronger upper mantle lower mantle viscosity contrasts than the cases where we 11 of 14

12 Figure 9. As in Figure 3, but for models (left) B1000f, (middle) B1000e, and (right) B1000c. found mantle planform responded minimally to boundary motion (i.e., contrasts greater than a factor of 100). 5. Conclusions [47] We investigated the influence of mobile divergent and convergent plate boundaries in mantle convection models with different ratios of upper mantle to lower mantle viscosity. For large viscosity contrasts the response of the lower mantle planform to mobile divergent plate boundaries (i.e., oceanic ridge-type boundaries) is minor, while the migration of convergent plate boundaries (i.e., analogous to subduction zones) has a strong influence on the lower mantle convective flow pattern. [48] Convection is strongly influenced by divergent plate boundary motion when the lower mantle is just 30 times more viscous than the upper mantle. In that case (model A30e), we found that the increasing mismatch between the plate boundary locations and mantle planform resulted in a plate reorganization event characterized by a rapid change in plate directions and speeds. Plate reorganization events were not observed in the cases featuring higher degrees of mantle viscosity stratification. We suggest this is because the coupling between the lower mantle planform and plate velocities is damped out as the upper mantle viscosity decreases relative to the lower mantle and plate viscosity. [49] Conversely, the motion of convergent plate boundaries affects the lower mantle even when the upper mantle is as much as 3 orders of magnitude less viscous than the lower mantle. In this case, plate boundary motion is communicated to the lower mantle directly through the thermal field, rather than the flow field, by the lateral migration of cold downwelling plate material. However, we find that a lower mantle that is much more viscous than the upper mantle does effect mobile downwellings quite differently from the downwellings found with fixed boundaries. As convergent plate boundaries move further from their initial location the associated downwellings can fail to entirely penetrate into the lower mantle or they can lose the sheet-like morphology exhibited in the cases with stationary plate boundaries. In a calculation with fixed plate boundary locations, entrainment of the upper mantle downwelling into the lower mantle by material descending in the high-viscosity lower mantle aids sinking upper mantle material to penetrate the lower mantle [Jarvis and Lowman, 2007]. When plate boundaries evolve, upper mantle downwellings no longer descend directly over lower mantle downwellings so that the lower mantle viscosity increase becomes a greater barrier to sinking upper mantle material. [50] Our findings indicate that plate boundary motion has a significant effect on the plate-mantle system and that models featuring stationary plate boundaries cannot fully capture the dynamics of a system like the Earth s mantle. For example, our results suggest that fixed plate geometries will exaggerate the vigor of downwellings and the mean surface heat flux from the convecting system. Mantle convection studies featuring evolving plate geometries as well as continual plate-like surface behavior [e.g., Tackley, 2000b, 12 of 14

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