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

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Earth and Planetary Science Letters

www.elsevier.com/locate/epsl

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:Received 13 July 2017Received in revised form 26 October 2017Accepted 9 December 2017Available online 3 January 2018Editor: B. Buffett

Keywords:passive margingrain damagesubduction initiationlithospheric rheology

The collapse of passive margins has been proposed as a possible mechanism for the spontaneous initiation of subduction. In order for a new trench to form at the junction between oceanic and continental plates, the cold and stiff oceanic lithosphere must be weakened sufficiently to deform at tectonic rates. Such rates are especially hard to attain in the cold ductile portion of the lithosphere, at which the mantle lithosphere reaches peak strength. The amount of weakening required for the lithosphere to deform in this tectonic setting is dictated by the available stress. Stress in a cooling passive margin increases with time (e.g., due to ridge push), and is augmented by stresses present in the lithosphere at the onset of rifting (e.g., due to drag from underlying mantle flow). Increasing stress has the potential to weaken the ductile portion of the lithosphere by dislocation creep, or by decreasing grain size in conjunction with a grain-size sensitive rheology like diffusion creep. While the increasing 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

Knowing how and why plate tectonics exists is critical for un-derstanding Earth’s surface and interior evolution, as well as that of other planets. The main driving force for plate motion ap-pears to come from the negative buoyancy of subducting slabs, as evident in the correlation between the fraction of subduction zone boundary per plate, and the plate’s velocity (Forsyth and Uyeda, 1975). However, the origin of subduction, and specifically the mechanism for its initiation remain enigmatic, because as the lithosphere becomes colder, denser and more likely to sink, it also

* Corresponding author.E-mail addresses: elvira.mulyukova@yale.edu (E. Mulyukova),

david.bercovici@yale.edu (D. Bercovici).

https://doi.org/10.1016/j.epsl.2017.12.0220012-821X/© 2017 Elsevier B.V. All rights reserved.

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 initiated during the Cenozoic, which indicates that subduction ini-tiation is a commonly occurring process on modern Earth (Stern, 2004). Moreover, it is reasonable to assume that spontaneous sub-duction occurred at some point in the Earth’s history, as part of the initiation of global plate tectonics. It is worth noting that there exist models for the onset of plate tectonics through other means, such as the lithospheric collapse at an active transform plate boundary (Casey and Dewey, 1984; Stern, 2004), formation of new plate boundaries by accumulation of lithospheric dam-age from proto-subduction (Bercovici and Ricard, 2014), or the plume-induced subduction initiation (Gerya et al., 2015). We ex-pect that as the physical conditions on an evolving Earth change (e.g., its mantle and surface temperature, continental cover, con-vective lengthscale, etc.), so does its style of plate tectonics, sub-

342 E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352

duction and subduction initiation. Whether it is possible to spon-taneously form a new trench at modern Earth conditions, or if modern subduction relies on the tectonic features inherited from 1–3 billions of years of ongoing plate tectonics, is currently not know.

The coincidence of many currently active subduction zones with the continental margins prompts the hypothesis that spon-taneous subduction initiation is caused by the collapse of passive margins (e.g., Stern, 2004), which is also invoked as the mecha-nism for ocean closure in the Wilson cycle (Wilson, 1968).

As new oceanic lithosphere forms after continental rifting, it cools and the stress at its bounding passive margin increases due to ridge push (Dahlen, 1981), thermal contraction (Korenaga, 2007), tensile stresses at the junction between the oceanic litho-sphere and the continent (Kemp and Stevenson, 1996), flexure under sedimentary loads (Cloetingh et al., 1989; Regenauer-Lieb et al., 2001), and possibly the stress due to drag from underlying mantle flow (Gurnis, 1988; Lenardic et al., 2011). Imposed and/or increasing stress has the potential to weaken the ductile portion of the lithosphere by dislocation creep, or by decreasing grain size in conjunction with a grain-size sensitive rheology like diffusion creep. The nonlinear coupling of viscosity to stress or to grain size can induce a self-weakening feedback (Montési and Hirth, 2003; Bercovici and Ricard, 2005; Rozel et al., 2011; Bercovici and Ricard, 2012), which manifests itself as a localized shear zone. However, this weakening is opposed by the thermally induced stiffening dur-ing lithospheric cooling, and the dominance of one effect or the other determines whether or not the margin might weaken and collapse, possibly allowing for the plate to subduct and form a new trench.

Here, we present a model of the thermal and mechanical evolu-tion of a passive margin, wherein we predict formation of a weak shear zone that spans a significant depth-range of the nominally cold and strong ductile portion of the lithosphere. The rates of cooling, grain growth and grain damage determine the rheologi-cal response and weakening of the margin through time. We posit that grain size reduction by two-phase grain-damage in the duc-tile portion of the lithosphere can enable, or at least significantly facilitate, the collapse of a passive margin. Applying these results to the expected conditions on early and modern Earth sheds light on when in Earth’s history the conditions may have been optimal for the initiation of subduction. For the thermal conditions repre-sentative of Venus, our model predicts that the rate of grain size reduction is too slow to offset thermal stiffening, precluding the formation of localized plate boundaries via the collapse of passive margins on that planet.

2. Stress and temperature evolution model

2.1. Stiffening due to cooling and how to offset it

As shown by Solomatov (1995), after Christensen (1984), cool-ing and thermal stiffening of the lithosphere should preclude plate tectonics and initiation of subduction. Thus, as in all problems of plate generation (Bercovici et al., 2015b), our goal is to understand what mechanisms and under which conditions offset thermal stiff-ening.

We limit our analysis to the depth-range of the passive mar-gin where viscous deformation is thought to dominate, which is, for 50 Myr old sea-floor, roughly from 10 to 80 km. This relatively cold and ductile region is considered to be the strongest part of the lithosphere (e.g., Kohlstedt et al., 1995) and is therefore a bot-tleneck for strain localization and plate boundary formation.

We assume that lithospheric rocks have a composite rheology, whereby the deformation can be accommodated by diffusion and dislocation creep:

e = Aτn + B

rmτ (1)

where e and τ are the square roots of the second invariants of the strain rate and stress tensors, respectively, n and m are the stress and grain size exponents, respectively (see Table 1), and ris a quantity proportional to grain size (see Section 3). The dislo-cation and diffusion creep compliances A and B , respectively, are temperature-dependent and follow an Arrhenius-type relation (see Table 1). The effect of pressure across the depth of the lithosphere is assumed to be significantly less than that of temperature and is therefore not considered, for simplicity.

When the lithosphere predominantly deforms by dislocation creep, its effective viscosity is

μdisl = 1

2Aτn−1 (2)

In diffusion creep, the effective viscosity is

μdiff = rm

2B(3)

If the lithosphere cools from the upper mantle temperature of 1500 K to the temperature of 800 K at the top of the ductile portion of the lithosphere (∼ 10 km depth), the dislocation creep viscosity would increase by a factor of 1016 (see Table 1 for rel-evant rheological properties). This thermal stiffening can be offset by an increase in stress by a factor of 108.

For the same temperature drop (from 1500 K to 800 K), the diffusion creep viscosity would increase by a factor of 109 (see Table 1), which can be offset by a decrease in grain size by a factor of 10−3.

Observational and experimental data, as well as theoretical models, suggest that lower temperature, or shallower depth, is as-sociated with higher stress (see Section 2.2) and smaller grain size (Linckens et al., 2015; Mulyukova and Bercovici, 2017). Both of these effects act to lower the viscosity and counteract the effect of thermal stiffening. Moreover, an increase in stress acts to lower the viscosity both in dislocation creep, as outlined above, and in diffu-sion creep, due to anti-correlation between grain size and stress (see Section 3).

While the above analysis demonstrates that grain size reduction in diffusion creep, or stress increase in dislocation creep can offset thermally induced stiffening, it remains to be shown if these mech-anisms are efficient enough at the conditions typical of passive margins to significantly weaken the lithosphere, and subsequently trigger subduction. In this study, we demonstrate that weakening of the lithosphere by grain damage can reduce its viscosity to val-ues below that of the underlying upper mantle and lead to, or at least facilitate, the collapse of the passive margin.

2.2. Sources of stress

2.2.1. Ridge pushAs oceanic lithosphere moves away from a spreading center it

cools and undergoes internal deformation and subsidence relative to its original position at the ridge. This deformation is driven by a vertical adjustment of loads as the system evolves towards isostatic equilibrium. After isostasy is reached, there remains a de-viatoric stress, known as ridge push, which compensates for the pressure head associated with the seafloor topography (see Tur-cotte and Schubert, 2014, pp. 281–282). Ridge push stress τrp is compressive in the horizontal direction (defined so that the hori-zontal normal stress τxx > 0) and tensile in the vertical direction (τzz < 0), where τrp = τxx = −τzz , while shear stress-components are assumed to be zero. The ridge push stress is given by Dahlen (1981):

E. Mulyukova, D. Bercovici / Earth and Planetary Science Letters 484 (2018) 341–352 343

Table 1Material and model properties.

Property Symbol Value / Definition Dimension

Surface tension γ 1 J m−2

Phase volume fraction φi φ1 = 0.4, φ2 = 0.6Phase distribution function1 η 3φ1φ2 ≈ 0.72

Dislocation creep2 edisl = Aτn

Activation energy Edisl 530 kJ mol−1

Prefactor A0 1.1 · 105 MPa−n s−1

Stress-Exponent n 3Compliance A A0exp(− Edisl

RT ) MPa−n s−1

Diffusion creep2 ediff = Br−mτ

Activation energy Ediff 300 kJ mol−1

Prefactor B0 13.6 μmm MPa−1 s−1

Grain Size Exponent m 3

Compliance3 B(

π2

)−mB0exp(− Ediff

RT ) μmm MPa−1 s−1

Grain growth4

Activation energy EG 300 kJ mol−1

Prefactor G0 2 · 104 μmp s−1

Exponent p 2Grain growth rate GG G0exp(− EG

RT ) μmp s−1

Interface coarsening5

Exponent q 4Interface coarsening rate G I

qp (μm)q−p GG

250 μmq s−1

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 Olivine grain-growth law from Karato (1989), using EG = 300 instead of 200 kJ mol−1.5 Interface coarsening law from Bercovici and Ricard (2013), based on the analysis done in Bercovici and Ricard

(2012).

Fig. 1. Schematic of the model of the cooling lithosphere. Coordinate axes x, z cen-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.

τrp = 1

2g

∞∫z

(ρ − ρM)dz (4)

where g is gravitational acceleration, z is depth relative to the sea floor (z is positive downward), ρ is the density within the litho-sphere, and ρM is the density of the underlying mantle (Fig. 1). As the lithosphere cools, its density relative to the ambient man-tle increases, which leads to an increase in τrp. For the half-space cooling model of the lithosphere, the depth- and time-dependence of temperature are given by Turcotte and Schubert (2014, pp. 153–155):

T = T M − �T erfc( z

2√

κt

)(5)

where erfc is the complimentary error function, κ is the ther-mal diffusivity, t is time (where t = 0 is the onset of rifting and plate generation) and �T = T M − T W is the temperature-difference between the surface (T W ) and the mantle at the bottom of the lithosphere (T M ).

The thermal density anomaly associated with (5) turns (4) into (Dahlen, 1981):

τrp = ρMα�T g ierfc( z

2√

κt

)√κt (6)

where α is the thermal expansivity and ierfc(x) = ∫ ∞x erfc(x)dx =

(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 dif-ference across the lithosphere. However, the increase in stress due to ridge push needed to offset thermal stiffening depends on the temperature difference across the lithosphere, but not on the abso-lute mantle temperature. These effects have important implications for applications to conditions on early Earth (see Section 4.4).

2.2.2. Other sources of stressIn addition to ridge push (τrp), there are other sources of stress

in the lithosphere, and these include stresses due to thermal con-traction (Korenaga, 2007), tensile stresses at the junction between the oceanic lithosphere and the continent (Kemp and Stevenson, 1996), compositionally induced negative buoyancy, such as from a chemically dense surface layer remaining after freezing of the magma ocean (Foley et al., 2014), and lithospheric flexure due to sedimentary overburden (Cloetingh et al., 1989; Regenauer-Lieb et al., 2001). The flexural stress can contribute several hundreds MPa of stress in the lithosphere, but only at shallow depths. In partic-ular, the elastic thickness of the oceanic lithosphere is estimated to be the distance from the surface to the ∼ 400–800 K isotherms (Wessel, 1992), corresponding to ∼ 10–40 km in a 100 Myr old lithosphere. The flexural stress is compressive only in the top half of that depth-region, which is where it adds to the effect of ridge push. In the deeper half, the flexural stress is tensile and acts to 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, the compressive stress due to flexure and augmentation of ridge push occurs only over 5–20 km depths, which is important for the brittle portion of the lithosphere, as others have inferred (e.g., Cloetingh et al., 1989; Regenauer-Lieb et al., 2001), but is of little effect on the deeper ductile regions, where the high lithospheric strength creates a bottle neck for deformation.

Finally, the underlying convecting mantle can also impart stresses to the lithosphere on the order of 100 MPa (Gurnis, 1988). In the early stages of rifting, it is likely that the sub-continental thermally insulated hotter mantle would upwell (Gurnis, 1988; Lenardic et al., 2011), giving rise to divergent flow beneath an initiating rift, which drags the overlying lithosphere away from the ridge and induces compressive stresses that augment ridge push stress.

Stress sources such as ridge push, thermal contraction and flex-ure due to sediment deposition are expected to be small at the early stages of rifting, and only become significant later, when the sea floor has cooled and subsided, and when the sediments have had time to accumulate. However, by the time these time-dependent stresses become significant, the shallower portion of the lithosphere is cold and stiff, and the capacity for the accu-mulated stress to induce deformation and grain damage is signifi-cantly reduced. In contrast, mantle drag can drive the deformation at the passive margin from the onset of rifting, when the material is at its hottest and weakest (and thus when the rates of me-chanical work and grain damage are high), since the geometry and strength of the imposed large scale mantle flow is not necessarily dependent on the age of a rift zone.

Although mantle drag acts only at the bottom of the litho-sphere, the induced stress can be assumed constant across its en-tire width, since the surface can be well approximated by a free slip boundary.

We account for the potential contribution to stress from mantle drag with a single additional stress τc, whose value is varied, and which we assume is compressive. The total stress in our model is thus:

τ = τrp(t) + τc (7)

where t is time since the onset of rifting. For simplicity, and given their potentially limited contribution to deformation, as discussed above, we do not include specific models for the other time-dependent stress sources.

3. Grain size evolution model

The grain size of lithospheric rocks determines the creep mech-anism by which they deform, more specifically whether the rock’s rheology is sensitive to stress (in dislocation creep) or grain size (in diffusion creep). For a given mineralogy, the grain-size evolution is modeled by the grain-damage model (Bercovici and Ricard, 2012) which accounts for the competition between coarsening (which

drives grain-growth) and deformation and damage (which drives grain-size reduction). Both processes depend on temperature. The detailed description of the two-phase grain-damage model has been presented in Bercovici and Ricard (2012, 2013, 2014); Mu-lyukova and Bercovici (2017) and is briefly outlined below.

Our modeled lithospheric material is composed of two phases: 40% pyroxene and 60% olivine. The phases are assumed to be well mixed on the grain size scale, causing the grains of each phase to be pinned by those of the other phase. (See Bercovici and Skemer, 2017, for a discussion of grain mixing and damage.) In this case, the grain size evolution is controlled by the evolution of the rough-ness r of the interface separating the two phases (the mean grain size R is given by R = r/

√hG , where hG ≈ π/2 for our choice

of phase volume fractions (see Bercovici and Ricard, 2013, 2014, 2016; Bercovici et al., 2015a). This approximation is known as the pinned state limit (Bercovici and Ricard, 2012).

We further assume that the two phases have the same com-posite rheology, whereby the deformation can be accommodated by diffusion and dislocation creep, as shown in (1). The grain size evolution is dictated by the evolution of the interface coarseness:

dr

dt= ηG I

qr(q−1)− f I r2

γ η� (8)

where η is the phase distribution function, G I and q are the rate and the exponent of interface coarsening, respectively, f I is the damage partitioning fraction (see (11)) and γ is the surface tension (see Table 1). The mechanical work rate � is defined as:

� = 2eτ = 2Aτn+1(

1 +( rF

r

)m)(9)

where we used the constitutive relation (1) and defined the field boundary grain size rF , which marks the transition from diffusion to dislocation creep dominated regimes, as:

rF =( B

Aτn−1

)1/m(10)

The solution of the full two-phase grain damage model (wherein there are three different evolution equations: for the grain sizes of the two phases and for the interface roughness) shows that while the grain size evolution might initially diverge from that of the in-terface roughness, they do eventually converge on each other once they cross into the diffusion creep regime (Bercovici and Ricard, 2012, Section 4.3). Moreover, evolution of the interface roughness predicted by the full two-phase model and that predicted by the pinned state model are nearly identical (Bercovici and Ricard, 2012, Appendix H.2). Therefore, we expect that the results obtained in this study, which uses the pinned state limit approximation, are representative of the solution to the full two-phase system.

The partitioning fraction f I is the fraction of mechanical work that goes into interface surface energy via interface damage (dis-tortion, rending, and grain mixing) and results in grain size re-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

Fig. 3. Partitioning fraction f I (11) as a function of temperature for three different values of exponent k (see legend).

that goes into interface damage ( f I ) can be different from that go-ing into the grain damage ( fG ). In the pinned state limit, however, damage to the interface translates directly into grain damage, and so only f I needs to be specified. There are few experimental and observational constraints on the values of f I and fG . By construc-tion, f I and fG must be smaller than one (Bercovici and Ricard, 2012). Comparing the experimentally deduced piezometers with those predicted by the single-phase grain-damage theory suggests that fG decreases with increasing temperature (Rozel et al., 2011; Mulyukova and Bercovici, 2017), and we thus assume that f I has a similar trend. Here we propose the following general temperature-dependent function for f I :

f I = f1

( f2

f1

)(T kM−T k)/(T k

M−T kW )

(11)

where T M and T W are mantle and surface temperatures (Fig. 1), respectively, f1 and f2 are prescribed minimum and maximum values, associated with the hottest and coldest temperatures, re-spectively, and the exponent k determines the variation of f I be-tween those values (Fig. 3).

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 disloca-tion 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 size and viscosity by several orders of magnitude over just a few million years (Fig. 4).

The steady state dr/dt = 0 is reached when the rates of grain growth and comminution balance each other, which is when the following relation holds:

1 = 2q

γ η2

f I A

G Irq+1τn+1

(1 +

( rF

r

)m)(12)

where we used (8)–(10). According to (12), the conditions for steady state depend on stress and temperature (i.e., through their dependence on f I , G I , rF , etc.). In particular, the steady state grain size tends to decrease with decreasing temperature and increas-ing stress (Mulyukova and Bercovici, 2017). Following the plunge, the grain size tracks its steady state value, provided the steady state grain size does not change faster because of cooling than the grain-size can adjust to it, as governed by (8).

We note that the temperature-dependence of the steady state grain size is sensitive to some of the material properties in our model, most notably the partitioning fraction f I (see Section 3) and the activation energies for grain growth (EG ) and diffusion creep (E B ). For example, E B − EG partially determines whether the steady state grain size increases or decreases with tempera-ture (see (12) and Mulyukova and Bercovici, 2017). We chose to use E B = EG for this study (see Table 1), which helps to reduce the temperature-dependence of the steady state grain size, and whose values are well within the range proposed by the published ex-perimental studies (Karato and Wu, 1993; Evans et al., 2001; Hirth 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. 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 · 10−3.

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, the initial grain size is r0 = 1 cm, T W = 285 K, T M = 1500 K, f1 = 10−12, f2 = 10−2 and k = 10.

In order to offset thermal stiffening, the rate of grain size re-duction needs to be sufficiently high relative to the rate of cool-ing, which can occur when the grain size plunges abruptly. The rate at which the grains get damaged and shrink is proportional to the mechanical work rate (9), which is in turn proportional to stress and temperature. In an aging lithosphere, the tempera-ture decreases at the same time as the stress increases (due to ridge push). These competing effects dictate the onset time for the plunge, which is delayed by cooling, but promoted by increasing stress. Thus, the grain size evolution path in a cooling lithosphere contains a window of opportunity, when the temperature is not yet too low and the stress is already high enough, for a plunge in grain size to occur. In the following, we explore the physical con-ditions for which lithospheric weakening due to rapid decrease in grain size can occur.

4.2. Grain size and viscosity evolution in an aging passive margin

When there is no significant change in grain size through time, the lithosphere only stiffens as it cools, in which case the con-ditions for collapse of a passive margin become increasingly less favorable (Cloetingh et al., 1989). Such is the case in our model when, for example, the stress τc is too low to induce a plunge in grain size: the margin cools and stiffens before any damage-induced weakening can take place (Fig. 5, left column). Increasing the contribution to stress from τc introduces a plunge in grain size for some depth-regions (Fig. 5, middle column). Specifically, the lithosphere becomes subdivided into two layers: a deeper one that undergoes a plunge in grain size, and a shallower one where no plunge occurs. After the plunge in grain-size, the very slow grain growth in the deep damaged region helps to keep it mechanically weaker than the rest of the lithosphere for hundreds of millions of years (Fig. 5).

The plunge in grain size in certain depth-regions of a passive margin is reflected in the viscosity evolution along the different isotherms. As an example, at τc = 10 MPa the viscosity of the litho-sphere deeper than the 900 K isotherm is reduced to the upper mantle value (∼ 1021 Pa s) after approximately 100 Myr (Fig. 5, middle column), while the viscosity above the 800 K isotherm, which is where we assume that the brittle failure can contribute to deformation, never drops below 1025 Pa s. The shallow regions where the modeled viscosity reaches extremely high values of

> 1040 MPa s are shown for completeness and point to the lim-its of our rheological model, where deformation mechanisms other than viscous creep, such as brittle failure, would be dominant.

The larger the stress in the passive margin, the wider is the depth-range of the lithosphere that is weakened by grain damage. For sufficiently high values of τc, the entire lithosphere becomes weakened due to a rapid plunge in grain size (Fig. 5, right col-umn). For τc = 50 MPa, the viscosity of the lithosphere deeper than the 900 K isotherm is consistently at or below the upper mantle value, from the very onset of rifting. The viscosity along the 800 K isotherm is smaller than what is obtained at τc = 10 MPa (Fig. 5, middle column), and viscosities of 1025 Pa s and above are first reached when the lithosphere has cooled to ∼ 750 K and be-low.

The amount of mechanical work that is needed to induce a plunge in grain size depends on the fraction of work that goes to-wards grain damage, which is controlled in our model by the parti-tioning fraction f I . The partitioning fraction has a minimum value f1 at the upper mantle temperature T M and a maximum value f2 at the surface temperature T W (see (11)). Partitioning of work into grain damage is most efficient at cold temperatures, which is in the later stages of the evolution of the passive margin. This is when ridge push τrp is a significant source of stress in our model: τrp surpasses 10 MPa in the top 50 km after ∼ 50 Myr, which is comparable to our range of values for τc. Thus, the value of f2 in-fluences the rate of grain damage done by the stress sources that are dominant after ∼ 10 Myr since the onset of rifting. A larger f2

leads to a more extensive depth-region of the lithosphere that ex-periences a plunge in grain size (Fig. 6, left column, see also Fig. 7).

At the early stages of rifting, the lithosphere is hot and the par-titioning of work into grain damage is inefficient, with f I being close to its minimum value f1. Because the viscosity μ at high temperatures can be low, the work rate � ∼ τ 2/μ can be high. At this early stage, while ridge push τrp is negligible, τc is finite and can induce a large amount of deformational work and there-fore damage. Thus, the rate of grain size reduction in a young passive margin is controlled by the values of f1 and τc. A larger f1 causes more grain size reduction at the early stages of rifting, which weakens the passive margin and causes it to remain weak for over a hundred Myr, due to the relatively slow grain healing 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 = 1 cm, 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 depth within the first few million years of rifting (as that would preclude the scenario of a strong plate bounded by (potentially) weak passive margins), which puts a constraint on the values of f1 and τc. Mantle drag alone can result in τc of up to 100 MPa, so the value of f1 needs to be small enough such that the grain size is not immediately driven down to a micron size. We find that f1 ≈10−12 fits well with this constraint. Alternatively, if τc drops after the initial rifting event, then the lithosphere trailing the initial rift will be less damaged, stronger and hence more plate-like.

The grains at greater depths and hotter temperatures shrink to a steady state size that is larger than that at colder temperatures, in agreement with the previously published theoretical studies and field observations (Linckens et al., 2015; Mulyukova and Bercovici, 2017). In our model, the correlation of grain size and temperature, as is observed in lithospheric shear zones, is more readily satisfied by younger passive margins (Fig. 5, top row).

4.3. Systematics of the grain size plunge

The predominant physical quantities that determine the rate of grain size reduction in our model, and thus whether and at what time and depth the plunge occurs, are the initial grain size r0, stress τ , partitioning fraction f I and the surface and mantle tem-peratures T W and T M , respectively (Fig. 7).

4.3.1. Depth-range of grain size plungeThe plunge in grain size occurs at shallower depths (meaning

that a wider depth-region of the lithosphere is weakened by grain damage) in models with smaller initial grain size, larger stress or hotter surface or mantle temperatures (Fig. 7). The effects of stress, partitioning fraction and upper mantle temperature are especially significant in this regard. With ridge push alone (τc ≤ 1 MPa), the plunge in grain size never occurs and the lithosphere continues to stiffen as it cools. Increasing the overall stress by increasing τcup to 10 MPa allows for portions of the lithosphere deeper than ∼ 50 km to undergo a plunge in grain size and weaken. Using τc ≥ 50 MPa allows for the entire lithosphere to be weakened. Por-tions of the lithosphere where a plunge in grain size has occurred remain weak for over 100 Myr, thanks to continued cooling, which

retards grain growth and thus healing, while also increasing τrp

(Fig. 7, B).The variation of f I with temperature, modeled through k in

(11), also has a significant effect on the extent of the lithosphere that gets weakened by grain damage. A larger k means that f I is close to its maximum value over a larger range of temperatures (Fig. 3), and thus a more extensive region where a plunge in grain size occurs (Fig. 7, D). When k < 10, the plunge in grain size rarely, if ever, occurs, which is why we use k = 10 for most of the cases.

An increase in upper mantle temperature leads to two effects: (i) increasing T M while keeping the surface temperature constant increases the temperature drop across the lithosphere, and thus the stress due to ridge push (see (6)) and the associated work rate; (ii) higher temperature means lower viscosity and thus a higher mechanical work rate for a given stress. Both effects lead to an 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 allows for the weakening by grain damage to occur at depths >35 km and > 25 km, respectively (instead of the > 50 km at the modern average upper mantle temperature of ∼ 1500 K) (Fig. 7, F).

Raising the surface temperature leads to a lower viscosity in the shallower regions of the lithosphere, and thus a higher rate of work and grain damage in those regions. While the ridge push stress is lower in this case, due to a smaller temperature difference across the lithosphere, the effect of lower viscosity dominates and leads to a more extensive depth-range where plunge in grain size and weakening occur (Fig. 7, E).

The viscosity in regions that experience a plunge in grain size drops significantly: the post-plunge viscosity μP is generally lower than 1020 Pa s (Fig. 7, A–F). With the proposed upper mantle vis-cosities ranging from μM ∼ 1018–1021 MPa (e.g., Karato and Wu, 1993), the weakening by grain damage makes viscous resistance in the lithosphere sufficiently low to induce, or at least facilitate, collapse of the passive margin. Notably, the viscosity across the en-tire depth-region that has undergone a plunge appears to be very weakly depth-dependent: the grain size variation is such that it offsets any thermally induced viscosity variations.

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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 for when grain size crosses the field boundary are shown. In all cases, f1 = 10−12. When not being varied, r0 = 1 cm, τc = 10 MPa, f2 = 10−2, k = 10, T W = 285 K and T M = 1500 K.

4.3.2. Onset of grain size plungeFor cases with Earth-like surface and mantle temperatures, the

latest the plunge occurs at any depth is less than 100 Myrs for all the tested values of r0, τc, f2 and k (Fig. 8). If the plunge in grain size does not occur within ∼ 100 Myr after the onset of cooling, it is unlikely to occur at all, as continued cooling makes the condi-tions for the plunge ever less favorable.

For the portions of the lithosphere that undergo a plunge in grain size, the grains in the hottest regions are the last to shrink. This is due to a lower value of f I and higher coarsening rate at higher temperatures, which makes the grain damage less efficient.

4.4. Case studies: early Earth and Venus

Our model of viscosity evolution in an aging lithosphere can be applied to the evolution of passive margins on early Earth and on Venus, if such exist. A number of caveats, however, need to be considered first. The process of rifting and the associated pas-

sive margins would inevitably be different on a younger Earth or Venus, compared to that on modern Earth, due to (i) smaller vol-umes or absence of continental crust and (ii) higher surface and mantle temperatures. In a scenario where there are no continents, we assume that there exist tectonic settings that are analogous to passive margins, i.e., junctions between a newly emplaced material and a preexisting surface layer (e.g., a primordial crust on early Earth, or the plateaux on Venus). The thermally insulating effect of the continents in our model is assumed to yield divergent flow of the mantle underlying the rift zone. The associated drag con-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 thus imply a smaller τc. The general effects of higher upper man-tle and surface temperatures were discussed in Section 4.3, and we look at two specific cases of varying T W and T M in this section.

Although Venus has no plate tectonics, its surface appears to be young and mobile, with possible, albeit controversial, evidence for subduction (Schubert and Sandwell, 1995), and elevated plateaux

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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 (Wieczorek, 2015); these conditions thus warrant consideration of the stability of its possible passive margins.

The Venusian surface is observed to have a temperature of ∼ 740 K (Zolotov, 2015), and we assume that its average upper mantle temperature is similar to that on modern Earth (although it could be up to 100 K hotter, depending on thermal history models (e.g., see Landuyt and Bercovici, 2009)). Thus, the temper-ature difference across the Venusian lithosphere is smaller than that on Earth. Given that the mantle flow rate, and thus convec-tive stress (Turcotte and Schubert, 2014, pp. 275–277), scales with the temperature difference that drives convection, we assume that the stress τc from mantle drag on Venus is smaller than that on Earth. Numerical modeling studies of mantle convection on Venus, constrained by the observed surface topography, volcanism and geoid, suggest that Venus does not have a low-viscosity astheno-sphere similar to that on Earth (Huang et al., 2013). In the absence of a weak sub-lithospheric channel, the convective stress acting on the Venusian lithosphere is thought to be lower than that on Earth (Höink et al., 2012), further motivating our choice of a low value for τc. Finally, the absence of continents on Venus limits the

amount of thermal insulation and the associated mantle drag act-ing beneath its rifting zones. We thus choose a small value for τc

when applying our model to study a Venus-like planet, namely, τc = 1 MPa.

The resulting conditions preclude the occurrence of a grain size plunge in Venus’ lithosphere, and its cooling passive margin be-comes increasingly stiff (Fig. 9, left column). Although Venus’ high surface temperature results in lower thermal stiffening, compared to that on Earth, the rate of grain size reduction is too low to offset it and form localized weak plate boundaries. However, as shown in Section 4.3 (Fig. 7), the occurrence of the plunge in grain size is sensitive to the mantle temperature, which for Venus is currently highly uncertain.

We next consider the evolution of passive margins in a younger Earth. Since the Earth’s mantle cools more slowly than the surface, we assume the Archean Earth had a surface temperature similar to that for present day, but had a hotter upper mantle temperature. Lithospheric stress τc from mantle drag was arguably smaller in the past, due to the lower viscosity of the upper mantle at higher temperatures, as well as smaller continental volumes to thermally 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). In all cases shown, the initial grain size is r0 = 1 cm, τc = 10 MPa, f1 = 10−12, f2 = 10−2 and k = 10.

ing. We thus use a more conservative value for τc = 10 MPa. The resulting conditions allow for the plunge in grain size to occur at depths > 20 km (Fig. 9, right column), which is a wider depth range than that obtained with modern Earth’s upper mantle tem-perature (depths > 50 km, Fig. 5, middle column). A more exten-sive region of the lithosphere in which the weakening due to grain size reduction occurs implies that the collapse of the passive mar-gins on Earth could have occurred more readily in the past. If this is true, then subduction initiation might have been more active in the Archean, and modern day subduction benefited from the re-sulting accumulation of weak zones (Bercovici and Ricard, 2014). While a larger portion of the lithosphere gets weakened by grain damage at hotter conditions, it takes longer for the plunge in grain size to occur (Fig. 8). Our modeling results thus imply that on hot-ter younger Earth any given oceanic plate is more likely to founder (eventually) than those on modern Earth. However, ancient oceanic plates would spend more time at the surface before undergoing sufficient weakening and foundering into the mantle, possibly re-sulting in larger plates and larger mantle convection cells. These results are consistent with the geological record of passive mar-gins, which shows that the lifespans of Precambrian margins were longer than those from the Phanerozoic (Bradley, 2008).

5. Discussion and conclusions

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. The conditions at which this occurs are dictated by the values of the work partitioning fraction f I and the lithospheric stress. For a plausible range of parameters, stresses due to ridge push and man-tle drag are sufficient to induce weakening by grain size reduction, even when using conservative values for stress induced by man-tle flow (< 10 MPa). The portion of the lithosphere that undergoes a plunge in grain size becomes less viscous than the underlying upper mantle, thus removing the viscous resistance that would otherwise preclude foundering of the oceanic lithosphere.

The separation of the lithosphere into a layer that undergoes a plunge in grain size and thus weakens, and a layer that remains coarse-grained and stiff, provides a new explanation for why the thermally induced thickening of the lithosphere, as inferred by the observed sea floor topography, deviates from the half-space cool-ing model after a few tens of millions years (Parsons and Sclater, 1977). The damaged and weakened deep portions of the litho-sphere could readily ‘peel off’ or drip away (see Elkins-Tanton, 2005; Paczkowski et al., 2012), leaving behind a stiff layer at the surface, whose relatively flat horizontal boundary is defined by the depth at which the initial grain size and stress conditions pre-cluded a plunge in grain size.

The plunge in grain size is promoted by smaller initial grain size, higher stress and hotter mantle and surface temperatures. At modern day surface and mantle temperatures, the plunge in grain size, and the induced collapse of the passive margin, occurs within the first 100 Myr of rifting, or does not occur at all. Many present-day passive margins are older than 100 Myr (Müller et al., 2008) and, according to our model, have missed their opportunity to get weakened by rapid grain size reduction. However, if the mantle underlying the nascent rifting zone was hotter than todays average upper mantle (e.g., due to the thermal insulation from the conti-nents), then our model predicts a deep damaged weak zone within todays passive margins, which can promote their collapse even af-ter 100 Myr.

The point in Earth’s history when spontaneous subduction initi-ation via collapse of a passive margin could occur depends on the temperature and stress conditions in the aging lithosphere, as well as on the presence of continents. In particular, whether passive margin collapse (facilitated by grain-damage) occurs in a geologi-cally plausible time depends on the surface and mantle tempera-tures, which govern the rate of lithospheric cooling and stiffening, as well as the ridge push force. The divergent flow of the ther-mally insulated hotter mantle beneath a newly forming rift zone gives rise to additional stress, without which it is difficult to get the passive margin to collapse, implying an important role of the continents in our model. Other studies have also demonstrated the effect of the continental cover on mantle dynamics, continental

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drift and break-up of supercontinents through rifting (e.g., Coltice et al., 2009; Heron and Lowman, 2011; Rolf et al., 2012). In addi-tion to the mantle drag resulting from the thermal blanketing, the continents act to increase the stress acting on the passive margins by inducing edge-driven flow in the underlying mantle (King and Anderson, 1998). Moreover, the stress supplied by the mantle onto the lithosphere will be focused on the margins due to the pres-ence of the continents (Rolf and Tackley, 2011). The contribution to stress from mantle drag is particularly important because it can induce deformation in the lithosphere already at the early stages of rifting. This induces a damage ‘pulse’ in a nascent passive margin, after which cooling and mounting ridge push stress keep the dam-aged margin from healing further. One may speculate that, after some time, the stress due to mantle drag is reduced (as the mantle thermally equilibrates), after which the damage to the lithosphere emanating from a ridge is reduced. This would cause a stronger plate to trail the greatly weakened passive margin, resulting in strong plates that are bounded by weak boundaries, characteris-tic of tectonic plates on Earth.

The emerging picture of the formation and foundering of oceanic lithosphere throughout Earth’s history, based on the results from our model, is as follows. The mechanically strong shallow portion of the lithosphere was thinner and less viscous in the past, with a larger portion of the lithosphere being weakened by grain damage, which would facilitate subduction initiation. The oceanic lithosphere spent more time at the surface prior to subduction, due to the delayed onset time for grain size plunge at hotter con-ditions, which serves as an argument for larger plates and larger scale convection cells on early Earth.

Future observational studies that can be helpful in testing the extent to which our model is representative of the evolution of passive margins on Earth can address the correlation between the age and temperature of the passive margins and its grain size. A thermomechanical model of the evolution of a passive margin, using stress-, temperature- and grain size-dependent rheologies, could test the plausibility of our simplified model of stress and temperature evolution in an aging lithosphere. Finally, using our model of the ductile portion of the lithosphere to understand the spontaneous subduction initiation via the collapse of a passive margin would require coupling it with the evolution of the brittle uppermost portion of the lithosphere. However, our relatively sim-ple model for passive margin collapse with grain-damage provides a guiding framework and testable predictions for future studies.

Acknowledgements

This work was supported by NSF Grants EAR-1344538 and EAR-1135382.

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