Collapse of Passive Margins by Lithospheric Damage and Plunging Grain Size ∗ Elvira Mulyukova , David Bercovici

Earth and Planetary Science Letters 484 (2018) 341–352

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Earth and Planetary Science Letters

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Collapse of passive margins by lithospheric damage and plunging grain size ∗ Elvira Mulyukova , David Bercovici

Yale University, Department of Geology & Geophysics, New Haven, CT, USA a r t i c l e i n f o a b s t r a c t

Article history: The collapse of passive margins has been proposed as a possible mechanism for the spontaneous Received 13 July 2017 initiation of subduction. In order for a new trench to form at the junction between oceanic and Received in revised form 26 October 2017 continental plates, the cold and stiff oceanic lithosphere must be weakened sufficiently to deform at Accepted 9 December 2017 tectonic rates. Such rates are especially hard to attain in the cold ductile portion of the lithosphere, Available online 3 January 2018 at which the mantle lithosphere reaches peak strength. The amount of weakening required for the Editor: B. Buffett lithosphere to deform in this tectonic setting is dictated by the available stress. Stress in a cooling Keywords: passive margin increases with time (e.g., due to ridge push), and is augmented by stresses present in passive margin the lithosphere at the onset of rifting (e.g., due to drag from underlying mantle flow). Increasing stress grain damage has the potential to weaken the ductile portion of the lithosphere by dislocation creep, or by decreasing subduction initiation grain size in conjunction with a grain-size sensitive rheology like diffusion creep. While the increasing lithospheric rheology stress acts to weaken the lithosphere, the decreasing temperature acts to stiffen it, and the dominance of one effect or the other determines whether the margin might weaken and collapse. Here, we present a model of the thermal and mechanical evolution of a passive margin, wherein we predict formation of a weak shear zone that spans a significant depth-range of the ductile portion of the lithosphere. Stiffening due to cooling is offset by weakening due to grain size reduction, driven by the combination of imposed stresses and grain damage. Weakening via grain damage is modest when ridge push is the only source of stress in the lithosphere, making the collapse of a passive margin unlikely in this scenario. However, adding even a small stress-contribution from mantle drag results in damage and weakening of a significantly larger portion of the lithosphere. We posit that rapid grain size reduction in the ductile portion of the lithosphere can enable, or at least significantly facilitate, the collapse of a passive margin and initiate a new subduction zone. We use this model to estimate the conditions for passive margin collapse for modern and ancient Earth, as well as for Venus. © 2017 Elsevier B.V. All rights reserved.

1. Introduction stiffens and requires more force to deform and founder into the mantle (Cloetingh et al., 1989; Solomatov, 1995). A large proportion of subduction zones that are active today Knowing how and why plate tectonics exists is critical for un- initiated during the Cenozoic, which indicates that subduction ini- derstanding Earth’s surface and interior evolution, as well as that tiation is a commonly occurring process on modern Earth (Stern, of other planets. The main driving force for plate motion ap- 2004). Moreover, it is reasonable to assume that spontaneous sub- pears to come from the negative buoyancy of subducting slabs, duction occurred at some point in the Earth’s history, as part as evident in the correlation between the fraction of subduction of the initiation of global plate tectonics. It is worth noting that zone boundary per plate, and the plate’s velocity (Forsyth and there exist models for the onset of plate tectonics through other Uyeda, 1975). However, the origin of subduction, and speciﬁcally means, such as the lithospheric collapse at an active transform the mechanism for its initiation remain enigmatic, because as the plate boundary (Casey and Dewey, 1984; Stern, 2004), formation lithosphere becomes colder, denser and more likely to sink, it also of new plate boundaries by accumulation of lithospheric damage from proto-subduction (Bercovici and Ricard, 2014), or the plume-induced subduction initiation (Gerya et al., 2015). We expect that as the physical conditions on an evolving Earth change * Corresponding author. E-mail addresses: [email protected] (E. Mulyukova), (e.g., its mantle and surface temperature, continental cover, con- [email protected] (D. Bercovici). vective lengthscale, etc.), so does its style of plate tectonics, sub- https://doi.org/10.1016/j.epsl.2017.12.022 0012-821X/© 2017 Elsevier B.V. All rights reserved. 342 E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352

B duction and subduction initiation. Whether it is possible to spon- e˙ = Aτ n + τ (1) taneously form a new trench at modern Earth conditions, or if rm modern subduction relies on the tectonic features inherited from where e˙ and τ are the square roots of the second invariants of 1–3 billions of years of ongoing plate tectonics, is currently not the strain rate and stress tensors, respectively, n and m are the know. stress and grain size exponents, respectively (see Table 1), and r The coincidence of many currently active subduction zones is a quantity proportional to grain size (see Section 3). The dislo- with the continental margins prompts the hypothesis that spon- cation and diffusion creep compliances A and B, respectively, are taneous subduction initiation is caused by the collapse of passive temperature-dependent and follow an Arrhenius-type relation (see margins (e.g., Stern, 2004), which is also invoked as the mecha- Table 1). The effect of pressure across the depth of the lithosphere nism for ocean closure in the Wilson cycle (Wilson, 1968). is assumed to be significantly less than that of temperature and is As new oceanic lithosphere forms after continental rifting, it therefore not considered, for simplicity. cools and the stress at its bounding passive margin increases When the lithosphere predominantly deforms by dislocation due to ridge push (Dahlen, 1981), thermal contraction (Korenaga, creep, its effective viscosity is 2007), tensile stresses at the junction between the oceanic lithosphere and the continent (Kemp and Stevenson, 1996), flexure = 1 μdisl − (2) under sedimentary loads (Cloetingh et al., 1989; Regenauer-Lieb 2Aτ n 1 et al., 2001), and possibly the stress due to drag from underlying In diffusion creep, the effective viscosity is mantle flow (Gurnis, 1988; Lenardic et al., 2011). Imposed and/or m increasing stress has the potential to weaken the ductile portion r μdiff = (3) of the lithosphere by dislocation creep, or by decreasing grain size 2B in conjunction with a grain-size sensitive rheology like diffusion If the lithosphere cools from the upper mantle temperature of creep. The nonlinear coupling of viscosity to stress or to grain size 1500 K to the temperature of 800 K at the top of the ductile can induce a self-weakening feedback (Montési and Hirth, 2003; portion of the lithosphere (∼ 10 km depth), the dislocation creep Bercovici and Ricard, 2005; Rozel et al., 2011; Bercovici and Ricard, viscosity would increase by a factor of 1016 (see Table 1 for rel- 2012), which manifests itself as a localized shear zone. However, evant rheological properties). This thermal stiffening can be offset this weakening is opposed by the thermally induced stiffening dur- by an increase in stress by a factor of 108. ing lithospheric cooling, and the dominance of one effect or the For the same temperature drop (from 1500 K to 800 K), the other determines whether or not the margin might weaken and diffusion creep viscosity would increase by a factor of 109 (see collapse, possibly allowing for the plate to subduct and form a new Table 1), which can be offset by a decrease in grain size by a factor − trench. of 10 3. Here, we present a model of the thermal and mechanical evolu- Observational and experimental data, as well as theoretical tion of a passive margin, wherein we predict formation of a weak models, suggest that lower temperature, or shallower depth, is as- shear zone that spans a significant depth-range of the nominally sociated with higher stress (see Section 2.2) and smaller grain size cold and strong ductile portion of the lithosphere. The rates of (Linckens et al., 2015; Mulyukova and Bercovici, 2017). Both of cooling, grain growth and grain damage determine the rheologi- these effects act to lower the viscosity and counteract the effect of cal response and weakening of the margin through time. We posit thermal stiffening. Moreover, an increase in stress acts to lower the that grain size reduction by two-phase grain-damage in the duc- viscosity both in dislocation creep, as outlined above, and in diffu- tile portion of the lithosphere can enable, or at least significantly sion creep, due to anti-correlation between grain size and stress facilitate, the collapse of a passive margin. Applying these results (see Section 3). to the expected conditions on early and modern Earth sheds light While the above analysis demonstrates that grain size reduction on when in Earth’s history the conditions may have been optimal in diffusion creep, or stress increase in dislocation creep can offset for the initiation of subduction. For the thermal conditions repre- thermally induced stiffening, it remains to be shown if these mech- sentative of Venus, our model predicts that the rate of grain size anisms are efficient enough at the conditions typical of passive reduction is too slow to offset thermal stiffening, precluding the margins to significantly weaken the lithosphere, and subsequently formation of localized plate boundaries via the collapse of passive trigger subduction. In this study, we demonstrate that weakening margins on that planet. of the lithosphere by grain damage can reduce its viscosity to values below that of the underlying upper mantle and lead to, or at 2. Stress and temperature evolution model least facilitate, the collapse of the passive margin.

2.1. Stiffening due to cooling and how to offset it 2.2. Sources of stress

As shown by Solomatov (1995), after Christensen (1984), cool- 2.2.1. Ridge push ing and thermal stiffening of the lithosphere should preclude plate As oceanic lithosphere moves away from a spreading center it tectonics and initiation of subduction. Thus, as in all problems of cools and undergoes internal deformation and subsidence relative plate generation (Bercovici et al., 2015b), our goal is to understand to its original position at the ridge. This deformation is driven what mechanisms and under which conditions offset thermal stiffening. by a vertical adjustment of loads as the system evolves towards We limit our analysis to the depth-range of the passive mar- isostatic equilibrium. After isostasy is reached, there remains a de- gin where viscous deformation is thought to dominate, which is, viatoric stress, known as ridge push, which compensates for the for 50 Myr old sea-floor, roughly from 10 to 80 km. This relatively pressure head associated with the seafloor topography (see Tur- cold and ductile region is considered to be the strongest part of cotte and Schubert, 2014, pp. 281–282). Ridge push stress τrp is the lithosphere (e.g., Kohlstedt et al., 1995) and is therefore a bot- compressive in the horizontal direction (defined so that the hori- tleneck for strain localization and plate boundary formation. zontal normal stress τxx > 0) and tensile in the vertical direction We assume that lithospheric rocks have a composite rheology, (τzz < 0), where τrp = τxx =−τzz, while shear stress-components whereby the deformation can be accommodated by diffusion and are assumed to be zero. The ridge push stress is given by Dahlen dislocation creep: (1981): E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352 343

Table 1 Material and model properties. Property Symbol Value / Deﬁnition Dimension − Surface tension γ 1Jm2 Phase volume fraction φi φ1 = 0.4, φ2 = 0.6 1 Phase distribution function η 3φ1φ2 ≈ 0.72 2 ˙ n Dislocation creep edisl = Aτ −1 Activation energy Edisl 530 kJ mol 5 −n −1 Prefactor A0 1.1 · 10 MPa s Stress-Exponent n 3 − Edisl −n −1 Compliance AA0exp( RT ) MPa s 2 ˙ −m Diffusion creep ediff = Br τ −1 Activation energy Ediff 300 kJ mol m −1 −1 Prefactor B0 13.6μmMPa s Grain Size Exponent m 3 −m 3 π − Ediff m −1 −1 Compliance B 2 B0exp( RT ) μm MPa s Grain growth4 −1 Activation energy EG 300 kJ mol 4 p −1 Prefactor G0 2 · 10 μm s Exponent p 2 − EG p −1 Grain growth rate GG G0exp( RT ) μm s Interface coarsening5 Exponent q 4 q q−p GG q −1 Interface coarsening rate G I p (μm) 250 μm s 1 Phase distribution function from Bercovici and Ricard (2012). 2 Our model values for rheological parameters are representative of experimentally determined olivine creep laws (Karato and Wu, 1993; Hirth and Kohlstedt, 2003). − 3 Factor (π/2) m in the deﬁnition of compliance accounts for using the interface roughness r instead of mean grain size in the constitutive law, as part of the pinned state limit assumption (see Section 3). 4 −1 Olivine grain-growth law from Karato (1989), using EG = 300 instead of 200 kJ mol . 5 Interface coarsening law from Bercovici and Ricard (2013), based on the analysis done in Bercovici and Ricard (2012).

The thermal density anomaly associated with (5) turns (4) into (Dahlen, 1981): z √ τrp = ρM αTgierfc √ κt (6) 2 κt ∞ where is the thermal expansivity and ierfc x = erfc x dx = √ α ( ) x ( ) (1/ π) exp(−x2) − xerfc(x) (see Fig. 2). The viscosity increase due to cooling depends on the absolute temperature of the ambient mantle and on the temperature difference across the lithosphere. However, the increase in stress due Fig. 1. Schematic of the model of the cooling lithosphere. Coordinate axes x, z cen- to ridge push needed to offset thermal stiffening depends on the tered on the crest of the ridge, as shown. Distance x from the ridge is proportional to the age of the passive margin through the spreading velocity. temperature difference across the lithosphere, but not on the absolute mantle temperature. These effects have important implications ∞ for applications to conditions on early Earth (see Section 4.4). 1 τrp = g (ρ − ρM )dz (4) 2 2.2.2. Other sources of stress z In addition to ridge push (τrp), there are other sources of stress where g is gravitational acceleration, z is depth relative to the sea in the lithosphere, and these include stresses due to thermal con- floor (z is positive downward), ρ is the density within the litho- traction (Korenaga, 2007), tensile stresses at the junction between sphere, and ρM is the density of the underlying mantle (Fig. 1). the oceanic lithosphere and the continent (Kemp and Stevenson, As the lithosphere cools, its density relative to the ambient man- 1996), compositionally induced negative buoyancy, such as from tle increases, which leads to an increase in τrp. For the half-space a chemically dense surface layer remaining after freezing of the cooling model of the lithosphere, the depth- and time-dependence magma ocean (Foley et al., 2014), and lithospheric flexure due to of temperature are given by Turcotte and Schubert (2014, pp. sedimentary overburden (Cloetingh et al., 1989; Regenauer-Lieb et 153–155): al., 2001). The flexural stress can contribute several hundreds MPa z of stress in the lithosphere, but only at shallow depths. In partic- T = T M − T erfc √ (5) ular, the elastic thickness of the oceanic lithosphere is estimated 2 κt to be the distance from the surface to the ∼ 400–800 K isotherms where erfc is the complimentary error function, κ is the ther- (Wessel, 1992), corresponding to ∼ 10–40 km in a 100 Myr old mal diffusivity, t is time (where t = 0is the onset of rifting and lithosphere. The flexural stress is compressive only in the top half plate generation) and T = T M − T W is the temperature-difference of that depth-region, which is where it adds to the effect of ridge between the surface (T W ) and the mantle at the bottom of the push. In the deeper half, the flexural stress is tensile and acts to lithosphere (T M ). offset ridge push, but is also where the lithosphere can act more 344 E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352

Fig. 2. Evolution of temperature (5) and stress due to ridge push (6) according to the half space cooling model. Solid black lines indicate isotherms from 400 to 1200 K at 100 K intervals. viscously than elastically, especially along warmer isotherms. Thus, drives grain-growth) and deformation and damage (which drives the compressive stress due to flexure and augmentation of ridge grain-size reduction). Both processes depend on temperature. The push occurs only over 5–20 km depths, which is important for detailed description of the two-phase grain-damage model has the brittle portion of the lithosphere, as others have inferred (e.g., been presented in Bercovici and Ricard (2012, 2013, 2014); Mu- Cloetingh et al., 1989; Regenauer-Lieb et al., 2001), but is of little lyukova and Bercovici (2017) and is briefly outlined below. effect on the deeper ductile regions, where the high lithospheric Our modeled lithospheric material is composed of two phases: strength creates a bottle neck for deformation. 40% pyroxene and 60% olivine. The phases are assumed to be well Finally, the underlying convecting mantle can also impart mixed on the grain size scale, causing the grains of each phase to stresses to the lithosphere on the order of 100 MPa (Gurnis, 1988). be pinned by those of the other phase. (See Bercovici and Skemer, In the early stages of rifting, it is likely that the sub-continental 2017, for a discussion of grain mixing and damage.) In this case, thermally insulated hotter mantle would upwell (Gurnis, 1988; the grain size evolution is controlled by the evolution of the rough- Lenardic et al., 2011), giving rise to divergent flow beneath an ness r of the interface separating√ the two phases (the mean grain initiating rift, which drags the overlying lithosphere away from the size R is given by R = r/ hG , where hG ≈ π/2for our choice ridge and induces compressive stresses that augment ridge push of phase volume fractions (see Bercovici and Ricard, 2013, 2014, stress. 2016; Bercovici et al., 2015a). This approximation is known as the Stress sources such as ridge push, thermal contraction and flex- pinned state limit (Bercovici and Ricard, 2012). ure due to sediment deposition are expected to be small at the We further assume that the two phases have the same com- early stages of rifting, and only become significant later, when posite rheology, whereby the deformation can be accommodated the sea floor has cooled and subsided, and when the sediments by diffusion and dislocation creep, as shown in (1). The grain size have had time to accumulate. However, by the time these time- evolution is dictated by the evolution of the interface coarseness: dependent stresses become significant, the shallower portion of 2 the lithosphere is cold and stiff, and the capacity for the accu- dr ηG I f I r = − (8) mulated stress to induce deformation and grain damage is signifi- dt qr(q−1) γη cantly reduced. In contrast, mantle drag can drive the deformation where is the phase distribution function, G and q are the rate at the passive margin from the onset of rifting, when the material η I is at its hottest and weakest (and thus when the rates of me- and the exponent of interface coarsening, respectively, f I is the chanical work and grain damage are high), since the geometry and damage partitioning fraction (see (11)) and γ is the surface tension strength of the imposed large scale mantle flow is not necessarily (see Table 1). The mechanical work rate is defined as: m dependent on the age of a rift zone. + rF = 2e˙τ = 2Aτ n 1 1 + (9) Although mantle drag acts only at the bottom of the litho- r sphere, the induced stress can be assumed constant across its en- where we used the constitutive relation (1) and defined the field tire width, since the surface can be well approximated by a free slip boundary. boundary grain size rF , which marks the transition from diffusion We account for the potential contribution to stress from mantle to dislocation creep dominated regimes, as: drag with a single additional stress τc, whose value is varied, and B 1/m rF = (10) which we assume is compressive. The total stress in our model is Aτ n−1 thus: The solution of the full two-phase grain damage model (wherein there are three different evolution equations: for the grain sizes of τ = τ (t) + τ (7) rp c the two phases and for the interface roughness) shows that while where t is time since the onset of rifting. For simplicity, and given the grain size evolution might initially diverge from that of the in- their potentially limited contribution to deformation, as discussed terface roughness, they do eventually converge on each other once above, we do not include specific models for the other time- they cross into the diffusion creep regime (Bercovici and Ricard, dependent stress sources. 2012, Section 4.3). Moreover, evolution of the interface roughness predicted by the full two-phase model and that predicted by the 3. Grain size evolution model pinned state model are nearly identical (Bercovici and Ricard, 2012, Appendix H.2). Therefore, we expect that the results obtained in The grain size of lithospheric rocks determines the creep mech- this study, which uses the pinned state limit approximation, are anism by which they deform, more specifically whether the rock’s representative of the solution to the full two-phase system. rheology is sensitive to stress (in dislocation creep) or grain size (in The partitioning fraction f I is the fraction of mechanical work diffusion creep). For a given mineralogy, the grain-size evolution is that goes into interface surface energy via interface damage (dis- modeled by the grain-damage model (Bercovici and Ricard, 2012) tortion, rending, and grain mixing) and results in grain size re- which accounts for the competition between coarsening (which duction. In a full two-phase damage model, the fraction of work E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352 345

4. Results

4.1. Grain size and viscosity evolution at constant temperature and stress

The rate of change in grain size is itself grain size dependent, as shown in (8). The work that goes into damaging the interface (right hand side of (8)), and which leads to grain size reduction, can be done by diffusion and dislocation creep. While in dislocation creep (i.e., r > rF ), the rate of damage is lower for smaller grains, and so the grains shrink more slowly as they get smaller. In contrast, the rate of damage in diffusion creep (i.e., r < rF ) is faster for smaller grains; this creates a strong positive feedback between the rate of deformation and grain size reduction, giving rise to self- weakening and causing the grain size to plunge rapidly towards its steady state. This abrupt plunge can involve a reduction in grain Fig. 3. Partitioning fraction f I (11) as a function of temperature for three different size and viscosity by several orders of magnitude over just a few values of exponent k (see legend). million years (Fig. 4). The steady state dr/dt = 0is reached when the rates of grain growth and comminution balance each other, which is when the following relation holds: that goes into interface damage ( f I ) can be different from that go- m ing into the grain damage ( f ). In the pinned state limit, however, 2q f I A + + rF G 1 = rq 1 n 1 1 + (12) 2 τ damage to the interface translates directly into grain damage, and γη G I r so only f needs to be speciﬁed. There are few experimental and I where we used (8)–(10). According to (12), the conditions for observational constraints on the values of f I and f G . By construc- steady state depend on stress and temperature (i.e., through their tion, f and f must be smaller than one (Bercovici and Ricard, I G dependence on f I , G I , rF , etc.). In particular, the steady state grain 2012). Comparing the experimentally deduced piezometers with size tends to decrease with decreasing temperature and increas- those predicted by the single-phase grain-damage theory suggests ing stress (Mulyukova and Bercovici, 2017). Following the plunge, that f G decreases with increasing temperature (Rozel et al., 2011; the grain size tracks its steady state value, provided the steady Mulyukova and Bercovici, 2017), and we thus assume that f I has a state grain size does not change faster because of cooling than the similar trend. Here we propose the following general temperature- grain-size can adjust to it, as governed by (8). We note that the temperature-dependence of the steady state dependent function for f I : grain size is sensitive to some of the material properties in our model, most notably the partitioning fraction f I (see Section 3) (T k −T k)/(T k −T k ) and the activation energies for grain growth (EG ) and diffusion f2 M M W f I = f1 (11) creep (E B ). For example, E B − EG partially determines whether f1 the steady state grain size increases or decreases with temperature (see (12) and Mulyukova and Bercovici, 2017). We chose to where T M and T W are mantle and surface temperatures (Fig. 1), use E B = EG for this study (see Table 1), which helps to reduce the respectively, f1 and f2 are prescribed minimum and maximum temperature-dependence of the steady state grain size, and whose values, associated with the hottest and coldest temperatures, re- values are well within the range proposed by the published ex- spectively, and the exponent k determines the variation of f I be- perimental studies (Karato and Wu, 1993; Evans et al., 2001; Hirth tween those values (Fig. 3). and Kohlstedt, 2003).

Fig. 4. Evolution of grain size (left) and viscosity (right) through time at constant stress and temperature (solid lines). Dashed lines indicate the ﬁeld boundary grain size. −3 Results for three different values of initial grain size are shown (see legend). For all models, we use T = 1100 K, τ = 50 MPa and f I = 5 · 10 . 346 E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352

Fig. 5. Evolution of grain size (top row) and viscosity (bottom row) in an aging lithosphere. Temperature evolves according to the half space cooling model (5), and stress according to the ridge push model (7). Solid black lines indicate isotherms from 400 to 1200 K at 100 K intervals. Results for different values of τc are shown. In all cases, −12 −2 the initial grain size is r0 = 1cm, T W = 285 K, T M = 1500 K, f1 = 10 , f2 = 10 and k = 10.

In order to offset thermal stiffening, the rate of grain size re- > 1040 MPa s are shown for completeness and point to the lim- duction needs to be sufficiently high relative to the rate of cool- its of our rheological model, where deformation mechanisms other ing, which can occur when the grain size plunges abruptly. The than viscous creep, such as brittle failure, would be dominant. rate at which the grains get damaged and shrink is proportional The larger the stress in the passive margin, the wider is the to the mechanical work rate (9), which is in turn proportional depth-range of the lithosphere that is weakened by grain damage. to stress and temperature. In an aging lithosphere, the tempera- For sufficiently high values of τc, the entire lithosphere becomes ture decreases at the same time as the stress increases (due to weakened due to a rapid plunge in grain size (Fig. 5, right col- ridge push). These competing effects dictate the onset time for the umn). For τc = 50 MPa, the viscosity of the lithosphere deeper plunge, which is delayed by cooling, but promoted by increasing than the 900 K isotherm is consistently at or below the upper stress. Thus, the grain size evolution path in a cooling lithosphere mantle value, from the very onset of rifting. The viscosity along the contains a window of opportunity, when the temperature is not 800 K isotherm is smaller than what is obtained at τc = 10 MPa yet too low and the stress is already high enough, for a plunge in (Fig. 5, middle column), and viscosities of 1025 Pa s and above are grain size to occur. In the following, we explore the physical con- first reached when the lithosphere has cooled to ∼ 750 K and be- ditions for which lithospheric weakening due to rapid decrease in low. grain size can occur. The amount of mechanical work that is needed to induce a plunge in grain size depends on the fraction of work that goes to- 4.2. Grain size and viscosity evolution in an aging passive margin wards grain damage, which is controlled in our model by the partitioning fraction f I . The partitioning fraction has a minimum value When there is no significant change in grain size through time, f1 at the upper mantle temperature T M and a maximum value the lithosphere only stiffens as it cools, in which case the con- f2 at the surface temperature T W (see (11)). Partitioning of work ditions for collapse of a passive margin become increasingly less into grain damage is most efficient at cold temperatures, which is favorable (Cloetingh et al., 1989). Such is the case in our model in the later stages of the evolution of the passive margin. This is when, for example, the stress τ is too low to induce a plunge c when ridge push τ is a significant source of stress in our model: in grain size: the margin cools and stiffens before any damage- rp τ surpasses 10 MPa in the top 50 km after ∼ 50 Myr, which is induced weakening can take place (Fig. 5, left column). Increasing rp comparable to our range of values for τc. Thus, the value of f2 in- the contribution to stress from τc introduces a plunge in grain size fluences the rate of grain damage done by the stress sources that for some depth-regions (Fig. 5, middle column). Specifically, the ∼ lithosphere becomes subdivided into two layers: a deeper one that are dominant after 10 Myr since the onset of rifting. A larger f2 undergoes a plunge in grain size, and a shallower one where no leads to a more extensive depth-region of the lithosphere that ex- plunge occurs. After the plunge in grain-size, the very slow grain periences a plunge in grain size (Fig. 6, left column, see also Fig. 7). growth in the deep damaged region helps to keep it mechanically At the early stages of rifting, the lithosphere is hot and the par- weaker than the rest of the lithosphere for hundreds of millions of titioning of work into grain damage is inefficient, with f I being years (Fig. 5). close to its minimum value f1. Because the viscosity μ at high ∼ 2 The plunge in grain size in certain depth-regions of a passive temperatures can be low, the work rate τ /μ can be high. margin is reflected in the viscosity evolution along the different At this early stage, while ridge push τrp is negligible, τc is finite isotherms. As an example, at τc = 10 MPa the viscosity of the litho- and can induce a large amount of deformational work and there- sphere deeper than the 900 K isotherm is reduced to the upper fore damage. Thus, the rate of grain size reduction in a young 21 mantle value (∼ 10 Pa s) after approximately 100 Myr (Fig. 5, passive margin is controlled by the values of f1 and τc. A larger middle column), while the viscosity above the 800 K isotherm, f1 causes more grain size reduction at the early stages of rifting, which is where we assume that the brittle failure can contribute which weakens the passive margin and causes it to remain weak to deformation, never drops below 1025 Pa s. The shallow regions for over a hundred Myr, due to the relatively slow grain healing where the modeled viscosity reaches extremely high values of rate (Fig. 6, left column). E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352 347

Fig. 6. Evolution of grain size (top row) and viscosity (bottom row) in an aging lithosphere. Temperature evolves according to the half space cooling model (5), and stress according to the ridge push model (6). Solid black lines indicate isotherms from 400 to 1200 K at 100 K intervals. Results for different values of the partitioning fraction (11) are shown (indicated at the top of each ﬁgure-column). In all cases shown, the initial grain size is r0 = 1cm, T W = 285 K, T M = 1500 K, τc = 10 MPa and k = 10.

It is unlikely that the lithosphere is mylonitized across its entire retards grain growth and thus healing, while also increasing τrp depth within the first few million years of rifting (as that would (Fig. 7, B). preclude the scenario of a strong plate bounded by (potentially) The variation of f I with temperature, modeled through k in weak passive margins), which puts a constraint on the values of (11), also has a significant effect on the extent of the lithosphere f1 and τc. Mantle drag alone can result in τc of up to 100 MPa, so that gets weakened by grain damage. A larger k means that f I is the value of f needs to be small enough such that the grain size is 1 close to its maximum value over a larger range of temperatures not immediately driven down to a micron size. We find that f1 ≈ −12 (Fig. 3), and thus a more extensive region where a plunge in grain 10 fits well with this constraint. Alternatively, if τc drops after size occurs (Fig. 7, D). When k < 10, the plunge in grain size rarely, the initial rifting event, then the lithosphere trailing the initial rift = will be less damaged, stronger and hence more plate-like. if ever, occurs, which is why we use k 10 for most of the cases. The grains at greater depths and hotter temperatures shrink to An increase in upper mantle temperature leads to two effects: a steady state size that is larger than that at colder temperatures, (i) increasing T M while keeping the surface temperature constant in agreement with the previously published theoretical studies and increases the temperature drop across the lithosphere, and thus field observations (Linckens et al., 2015; Mulyukova and Bercovici, the stress due to ridge push (see (6)) and the associated work rate; 2017). In our model, the correlation of grain size and temperature, (ii) higher temperature means lower viscosity and thus a higher as is observed in lithospheric shear zones, is more readily satisfied mechanical work rate for a given stress. Both effects lead to an by younger passive margins (Fig. 5, top row). increased rate of grain damage and to more extensive regions that experience a plunge in grain size. Increasing T M by 100 or 200 K 4.3. Systematics of the grain size plunge allows for the weakening by grain damage to occur at depths > 35 km and > 25 km, respectively (instead of the > 50 km at the The predominant physical quantities that determine the rate of modern average upper mantle temperature of ∼ 1500 K) (Fig. 7, F). grain size reduction in our model, and thus whether and at what Raising the surface temperature leads to a lower viscosity in time and depth the plunge occurs, are the initial grain size r0, the shallower regions of the lithosphere, and thus a higher rate stress τ , partitioning fraction f I and the surface and mantle temperatures T W and T M , respectively (Fig. 7). of work and grain damage in those regions. While the ridge push stress is lower in this case, due to a smaller temperature difference 4.3.1. Depth-range of grain size plunge across the lithosphere, the effect of lower viscosity dominates and The plunge in grain size occurs at shallower depths (meaning leads to a more extensive depth-range where plunge in grain size that a wider depth-region of the lithosphere is weakened by grain and weakening occur (Fig. 7, E). damage) in models with smaller initial grain size, larger stress or The viscosity in regions that experience a plunge in grain size hotter surface or mantle temperatures (Fig. 7). The effects of stress, drops significantly: the post-plunge viscosity μP is generally lower partitioning fraction and upper mantle temperature are especially than 1020 Pa s (Fig. 7, A–F). With the proposed upper mantle vis- significant in this regard. With ridge push alone (τ ≤ 1MPa), the c cosities ranging from μ ∼ 1018–1021 MPa (e.g., Karato and Wu, plunge in grain size never occurs and the lithosphere continues M 1993), the weakening by grain damage makes viscous resistance to stiffen as it cools. Increasing the overall stress by increasing τc up to 10 MPa allows for portions of the lithosphere deeper than in the lithosphere sufficiently low to induce, or at least facilitate, ∼ 50 km to undergo a plunge in grain size and weaken. Using collapse of the passive margin. Notably, the viscosity across the en- τc ≥ 50 MPa allows for the entire lithosphere to be weakened. Por- tire depth-region that has undergone a plunge appears to be very tions of the lithosphere where a plunge in grain size has occurred weakly depth-dependent: the grain size variation is such that it remain weak for over 100 Myr, thanks to continued cooling, which offsets any thermally induced viscosity variations. 348 E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352

Fig. 7. The minimum viscosity reached after a plunge in grain size, deﬁned as the minimum viscosity value from the time-period when the material is simultaneously in the diffusion creep regime and undergoes grain size reduction. Results are shown for different values of (A) initial grain size r0, (B) stress τc (acting in addition ridge push), (C) maximum damage partitioning fraction f2 (see (11)), (D) upper mantle temperature T M , (E) surface temperature T W , and (F) temperature exponent k of the partitioning fraction (see (11)). Colors of the markers indicate different depths (see colorbar). Square markers indicate cases where a plunge did not occur; in these cases the viscosity −12 −2 for when grain size crosses the ﬁeld boundary are shown. In all cases, f1 = 10 . When not being varied, r0 = 1cm, τc = 10 MPa, f2 = 10 , k = 10, T W = 285 K and T M = 1500 K.

4.3.2. Onset of grain size plunge sive margins would inevitably be different on a younger Earth or For cases with Earth-like surface and mantle temperatures, the Venus, compared to that on modern Earth, due to (i) smaller vol- latest the plunge occurs at any depth is less than 100 Myrs for all umes or absence of continental crust and (ii) higher surface and the tested values of r0, τc, f2 and k (Fig. 8). If the plunge in grain mantle temperatures. In a scenario where there are no continents, size does not occur within ∼ 100 Myr after the onset of cooling, it we assume that there exist tectonic settings that are analogous to is unlikely to occur at all, as continued cooling makes the condi- passive margins, i.e., junctions between a newly emplaced material tions for the plunge ever less favorable. and a preexisting surface layer (e.g., a primordial crust on early For the portions of the lithosphere that undergo a plunge in Earth, or the plateaux on Venus). The thermally insulating effect grain size, the grains in the hottest regions are the last to shrink. of the continents in our model is assumed to yield divergent flow This is due to a lower value of f I and higher coarsening rate at of the mantle underlying the rift zone. The associated drag con- higher temperatures, which makes the grain damage less efficient. tributes to τc, which can be up to 100 MPa on Earth (Gurnis, 1988; Lenardic et al., 2011). The absence of continents on Venus would 4.4. Case studies: early Earth and Venus thus imply a smaller τc. The general effects of higher upper mantle and surface temperatures were discussed in Section 4.3, and we Our model of viscosity evolution in an aging lithosphere can look at two specific cases of varying T W and T M in this section. be applied to the evolution of passive margins on early Earth and Although Venus has no plate tectonics, its surface appears to be on Venus, if such exist. A number of caveats, however, need to young and mobile, with possible, albeit controversial, evidence for be considered first. The process of rifting and the associated pas- subduction (Schubert and Sandwell, 1995), and elevated plateaux E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352 349

Fig. 8. Same as Fig. 7, but showing the onset time of the plunge in grain size (only for the cases where the plunge occurs), which is defined as the time at which the viscosity hits a minimum value while the material is in diffusion creep and experiencing grain size reduction. like Ishtar and Aphrodite that possibly act like continental crust amount of thermal insulation and the associated mantle drag act- (Wieczorek, 2015); these conditions thus warrant consideration of ing beneath its rifting zones. We thus choose a small value for τc the stability of its possible passive margins. when applying our model to study a Venus-like planet, namely, The Venusian surface is observed to have a temperature of τc = 1MPa. ∼ 740 K (Zolotov, 2015), and we assume that its average upper The resulting conditions preclude the occurrence of a grain size mantle temperature is similar to that on modern Earth (although plunge in Venus’ lithosphere, and its cooling passive margin be- it could be up to 100 K hotter, depending on thermal history comes increasingly stiff (Fig. 9, left column). Although Venus’ high models (e.g., see Landuyt and Bercovici, 2009)). Thus, the temper- surface temperature results in lower thermal stiffening, compared ature difference across the Venusian lithosphere is smaller than to that on Earth, the rate of grain size reduction is too low to offset that on Earth. Given that the mantle flow rate, and thus convec- it and form localized weak plate boundaries. However, as shown tive stress (Turcotte and Schubert, 2014, pp. 275–277), scales with in Section 4.3 (Fig. 7), the occurrence of the plunge in grain size is the temperature difference that drives convection, we assume that sensitive to the mantle temperature, which for Venus is currently the stress τc from mantle drag on Venus is smaller than that on highly uncertain. Earth. Numerical modeling studies of mantle convection on Venus, We next consider the evolution of passive margins in a younger constrained by the observed surface topography, volcanism and Earth. Since the Earth’s mantle cools more slowly than the surface, geoid, suggest that Venus does not have a low-viscosity astheno- we assume the Archean Earth had a surface temperature similar to sphere similar to that on Earth (Huang et al., 2013). In the absence that for present day, but had a hotter upper mantle temperature. of a weak sub-lithospheric channel, the convective stress acting Lithospheric stress τc from mantle drag was arguably smaller in on the Venusian lithosphere is thought to be lower than that on the past, due to the lower viscosity of the upper mantle at higher Earth (Höink et al., 2012), further motivating our choice of a low temperatures, as well as smaller continental volumes to thermally value for τc. Finally, the absence of continents on Venus limits the insulate the mantle and induce divergent flow and drag upon rift- 350 E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352

Fig. 9. Same as Fig. 5, but for a hotter surface (T W = 740 K, left) or mantle (T M = 1600 K, right) temperatures, compared to the modern Earth (T W = 285 K and T M = 1500 K). −12 −2 In all cases shown, the initial grain size is r0 = 1cm, τc = 10 MPa, f1 = 10 , f2 = 10 and k = 10. ing. We thus use a more conservative value for τc = 10 MPa. The The separation of the lithosphere into a layer that undergoes a resulting conditions allow for the plunge in grain size to occur at plunge in grain size and thus weakens, and a layer that remains depths > 20 km (Fig. 9, right column), which is a wider depth coarse-grained and stiff, provides a new explanation for why the range than that obtained with modern Earth’s upper mantle tem- thermally induced thickening of the lithosphere, as inferred by the perature (depths > 50 km, Fig. 5, middle column). A more exten- observed sea floor topography, deviates from the half-space cool- sive region of the lithosphere in which the weakening due to grain ing model after a few tens of millions years (Parsons and Sclater, size reduction occurs implies that the collapse of the passive mar- 1977). The damaged and weakened deep portions of the litho- gins on Earth could have occurred more readily in the past. If this sphere could readily ‘peel off’ or drip away (see Elkins-Tanton, is true, then subduction initiation might have been more active in 2005; Paczkowski et al., 2012), leaving behind a stiff layer at the the Archean, and modern day subduction benefited from the re- surface, whose relatively flat horizontal boundary is defined by the sulting accumulation of weak zones (Bercovici and Ricard, 2014). depth at which the initial grain size and stress conditions pre- While a larger portion of the lithosphere gets weakened by grain cluded a plunge in grain size. damage at hotter conditions, it takes longer for the plunge in grain The plunge in grain size is promoted by smaller initial grain size to occur (Fig. 8). Our modeling results thus imply that on hot- size, higher stress and hotter mantle and surface temperatures. At ter younger Earth any given oceanic plate is more likely to founder modern day surface and mantle temperatures, the plunge in grain (eventually) than those on modern Earth. However, ancient oceanic size, and the induced collapse of the passive margin, occurs within plates would spend more time at the surface before undergoing the first 100 Myr of rifting, or does not occur at all. Many present- sufficient weakening and foundering into the mantle, possibly re- day passive margins are older than 100 Myr (Müller et al., 2008) sulting in larger plates and larger mantle convection cells. These and, according to our model, have missed their opportunity to get results are consistent with the geological record of passive mar- weakened by rapid grain size reduction. However, if the mantle gins, which shows that the lifespans of Precambrian margins were underlying the nascent rifting zone was hotter than todays average longer than those from the Phanerozoic (Bradley, 2008). upper mantle (e.g., due to the thermal insulation from the continents), then our model predicts a deep damaged weak zone within todays passive margins, which can promote their collapse even af- 5. Discussion and conclusions ter 100 Myr. The point in Earth’s history when spontaneous subduction initi- We posit that rapid grain size reduction in the ductile portion ation via collapse of a passive margin could occur depends on the of the lithosphere can enable, or at least significantly facilitate, the temperature and stress conditions in the aging lithosphere, as well collapse of a passive margin and initiate a new subduction zone. as on the presence of continents. In particular, whether passive The conditions at which this occurs are dictated by the values of margin collapse (facilitated by grain-damage) occurs in a geologi- the work partitioning fraction f I and the lithospheric stress. For a cally plausible time depends on the surface and mantle tempera- plausible range of parameters, stresses due to ridge push and man- tures, which govern the rate of lithospheric cooling and stiffening, tle drag are sufficient to induce weakening by grain size reduction, as well as the ridge push force. The divergent flow of the ther- even when using conservative values for stress induced by man- mally insulated hotter mantle beneath a newly forming rift zone tle flow (< 10 MPa). The portion of the lithosphere that undergoes gives rise to additional stress, without which it is difficult to get a plunge in grain size becomes less viscous than the underlying the passive margin to collapse, implying an important role of the upper mantle, thus removing the viscous resistance that would continents in our model. Other studies have also demonstrated the otherwise preclude foundering of the oceanic lithosphere. effect of the continental cover on mantle dynamics, continental E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352 351 drift and break-up of supercontinents through rifting (e.g., Coltice Bradley, D.C., 2008. Passive margins through earth history. Earth-Sci. Rev. 91 (1), et al., 2009; Heron and Lowman, 2011; Rolf et al., 2012). In addi- 1–26. http://www.sciencedirect.com/science/article/pii/S0012825208000871. tion to the mantle drag resulting from the thermal blanketing, the Casey, J.F., Dewey, J.F., 1984. 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