Here, we test first the ability of small-scale convection to initiate subduction with the two different top boundary conditions and, second, the importance of basal heating and an initial stagnant-lid phase on the mode of plate tectonics in global models of mantle convection.
Subduction initiation by small-scale convection
A lithosphere below a free surface can be expected to develop a different stress pattern (e.g. through slight convection-induced bending) than a lithosphere that is vertically fixed at its top (Thielmann et al. 2015). We, therefore, compare the behaviour of an intact, stagnant lid that is either vertically fixed or can freely evolve in the vertical direction. Solomatov (2004) provides a numerical study of such a setup (using a vertically-fixed, free-slip top boundary) that investigates the ability of stresses exerted by sub-lithospheric small-scale convection to break a stagnant lid. Some of these models are reproduced and compared to additional models with a free surface.
No yield stress
Deformation in Earth’s mantle is mainly controlled by its temperature- and pressure-dependent rheology. Yet, it is now an established fact that a temperature- and pressure-dependent rheology alone leads to a flowing mantle with a stagnant lid on its top (Nataf and Richter 1982; Ratcliff et al. 1997; Solomatov 1995; Stengel et al. 1982). The cold, stagnant top boundary layer is thereby too strong to be broken apart by the stresses exerted via the coupling to the convecting, low-viscosity mantle underneath: Subduction does not initiate. The convective stresses are thereby focussed at the uppermost part of the lid, its coldest and thus strongest part (see Fig. 2
a and Additional file 1: Figure S1.1). A non-Newtonian weakening mechanism like brittle or ductile yielding has previously been shown to be the key mechanism to prevent a stagnant lid and to allow for subduction (Tackley 2000a).
For the study on subduction initiation by small-scale convection enabled by plastic yielding, we therefore present three suites of a simple regional model with either a simple depth-constant yield stress, a depth-dependent yield stress or a more realistic, composite yield stress by superimposing the two previous formulations (see Fig. 1).
Depth-constant yield stress
Solomatov (2004) employed experiments with varying plate strength that tested different constant (i.e. depth-independent) yield stresses. In one set of experiments, the lithosphere weakened solely by employing a (non-dimensional) depth-independent yield stress σ
y,const. He found that small-scale convection is able to initiate subduction in weak plates with σ
y,const=0.0003 (4.4 MPa) but not in “strong” plates with σ
y,const=0.0004 (5.8 MPa). This result is confirmed by our models shown in Fig. 3
a–c.
Interestingly, subduction of a “strong” plate with σ
y,const=0.0004 (5.8 MPa) can be initiated using the same model setup plus a free surface (Fig. 3
e). The time it takes for subduction to initiate is longer than in the case with a lower σ
y,const, where subduction occurs more readily (Fig. 3
d). An even stronger plate with σ
y,const=0.0005 (7.3 MPa), however, can resist stresses exerted by small-scale convection even with a free surface and does not result in subduction initiation (Fig. 3
f).
Depth-dependent yield stress
In a second suite of experiments, we consider a Byerlee-type, depth-dependent yield stress, again according to Solomatov (2004). The minimum variation of the Δ
σ
y
parameter between two different experiments is as low as possible to still be resolved spatially: The vertical resolution at lithospheric depths is around 10 km, which still resolves a change of the yield stress gradient (Δ
σ
y
) of ±0.001 at around 100 km depth (i.e. at the lithosphere-asthenosphere boundary) in the parameter range chosen (see Fig. 1). This is important to consider because the numerical grid slightly differs between the two models due to the missing/presence of the sticky-air layer.
Similarly to the depth-constant yield stress experiments, this comparison indicates that a free surface favours subduction initiation in comparison to a vertically fixed (free-slip) surface (see Table 2 and Additional file 1: Figures S1.3 and S1.4). Experiments with a vertically fixed surface develop subduction only until a yield stress gradient of Δ
σ
y
=0.0003 but not for higher values. Higher values of up to Δ
σ
y
=0.0005, however, still yield subduction in the free surface experiments. A yield stress gradient of Δ
σ
y
=0.0006 finally prevents subduction caused by small-scale convection in both models.
Composite yield stress
The third suite of experiments shown in Fig. 4 with the regional model applies a more Earth-like composite yield stress, with a brittle, depth-dependent yield stress in the upper part and a ductile, depth-constant part in the lower part of the lithosphere. The cross-over between brittle and ductile yielding is set such that it occurs approximately in the middle of the lithosphere, which is here set at 50 km depth.
Again, subduction occurs more readily when the vertically fixed surface is replaced by a free surface (Fig. 4
e): The maximum ductile yield stress that still allows subduction to occur is σ
y,const=0.0005 (7.3 MPa) and σ
y,const=0.00065 (9.5 MPa) for the free-slip and the free surface model, respectively. Limiting the strength of the lithosphere at shallow depth therefore allows for a 64 % stronger lower part that still entirely yields and subsequently subducts.
Free slip versus free surface
Modelling of mantle convection becomes more realistic when the top surface of the mantle is treated as a free surface instead of being vertically fixed (i.e. free slip): A free surface allows for natural plate bending (Crameri et al. 2012b). The plate bending in turn causes a characteristic lithospheric stress pattern resulting in two thin (instead of one thick) horizontal layers of high stress (Crameri 2013; Thielmann et al. 2015).
In our high-resolution regional models, we find that another striking difference between free-slip and free surface models exists. Both fixed and a free surface model setups lead to lithosphere-scale shear zones that, given a suitable flow rheology, eventually trigger subduction initiation. The most striking difference lies, however, within these shear zones: While a vertically fixed surface leads to broad, diffuse shear zones (Fig. 5
d), multiple criss-crossing, narrow shear zones next to each other characterise models with a free surface (Fig. 5
i). This striking difference can be attributed to the deformation in the uppermost parts of the plate. If the surface is fixed, vertical deformation at shallow depth has to be translated entirely into horizontal deformation due to mass conservation. A vertically, or near-vertically dipping shear zone (e.g. at 45°) therefore always involves a significant amount of horizontal strain, which causes the localised deformation to widen horizontally.
The free surface, on the contrary, enables a lithosphere-scale shear zone to maintain its strain direction throughout the plate interior to the plate surface by creating actual topography there. Since a single fault is rarely able to accommodate the total horizontal deformation of the plate, a characteristic crisscrossing pattern of aligned and mirrored shear zones dipping at around 45° forms as is visible in Fig. 5
g, i. This feature occurs throughout all the free-surface experiments considered here independent of applied yielding formulation, their time-dependent evolution or the numerical resolution (see Additional file 1: Figures S1.5 and S1.6). Additional figures and a movie show this in more detail (see Additional file 1: Figures S1.2 and S1.3, and the ). Small-scale horst and graben-like structures are thereby the topographic result of multiple normal or reverse faults situated next to each other in regions of extension or compression, respectively (Fig. 5
f). In contrast to the broad, diffuse shear zones in the free-slip surface models, straight and localised deformation zones occurring in models with a free surface are more efficient to take up tectonic stresses and allow the shear zones to reach deeper. A shear zone that reaches the bottom of the lithosphere is a prerequisite for subduction initiation as has been pointed out by Fowler and O’Brien (2003). A model with a free surface might therefore favour subduction initiation as discussed below (see Fig. 6).
Additional file 2: Movie of the experiment “lidC5” showing the temporal evolution of the regional model for (a) topography, (b) viscosity, the second invariant of (c) the stress and (d) the strain-rate tensor, (e) horizontal velocity and (f) the dominating deformation mechanism. (MOV 7114 kb)
Another mechanism to cause deeper brittle yielding might be the slight time-dependent bending of the plates while floating on a weaker, convecting mantle. The time-dependent mantle up- and downwellings induce a time-dependent vertical forcing on the plate floating above, which causes topography at the lithosphere-asthenosphere boundary (LAB). In the model where the top surface is vertically fixed, this deflection at the base of the plate is accommodated by plate-internal deformation. Releasing the plate surface from its vertical fixity, however, causes the plate to deflect in its entirety: positive and negative surface deflections matching the LAB topography occur above mantle up- and downwellings, respectively (Fig. 5
f). The strong plate-internal deformation occurring in the model with a vertically fixed surface is reduced when applying a free surface and accommodated by vertical deflection of the whole plate: The plate is bending.
While the convective stresses in the free-slip surface model cause internal deformation and convert to rather shallow accumulation of lithospheric stresses, they cause plate bending in the free surface model, which in turn increases shallow but also deep lithospheric stresses. Although the thermal thickness of the lid is similar for both upper boundary conditions, the high stress pattern in the case of the composite yield stress experiments reaches deeper in the case of a free surface (Fig. 5
c, h). Figure 2 shows the radial profiles of the root-mean-square (RMS) values of temperature, viscosity and the second invariant of the stress tensor for all models. The different stress distribution—solely caused by the free surface—is significant in all experiments, even though temperature and viscosity profiles are similar.
Plate bending favours another important tectonic mechanism: The lateral gradient in vertical plate deflection (i.e. tilting) causes a gravitation-induced, horizontal sliding of the plate segments. The plate segments, separated by the narrow weak zones of high deformation, tend to flow from the high pressure above mantle upwellings to the low pressure above mantle downwellings (Fig. 5
e, j). The tilting of the plate segments together with the resulting gravitational sliding cause regionally high lithospheric stresses (in between tilted segments) that lead to increased and deeper yielding in the case with a free surface. In the models with a fixed surface, mantle up- and downwellings cause lateral pressure gradients inside the plate that lead to horizontal flow similar to the gravitational sliding mentioned above. However, the plate segments are, due to the fixed top, not tilted and this does therefore not cause any additional stress.
The maximum depth of yielding can be taken as an indicator for lithosphere-scale shear zones. Figure 6 shows the maximum depth of brittle and/or ductile yielding during the temporal evolution of the models. The maximum depth of yielding increases shortly after the model’s initial state and during onset of convection in the mantle, while the plate thickness slowly decreases over time due to slight heating up of the model and subsequent erosion at the base of the plate. The different time-dependent fluctuation of the maximum yielding depth between free-slip and free surface models is mostly caused by the different top boundary conditions (and their implications) rather than variations in mantle flow, as the time-dependent evolution of the flow below the plate is similar for both model setups. Although the yielding occurs then throughout most of the thermal lithosphere in the free-slip model at around 3.8 Ga (see Fig. 6
a), it does not exceed its maximum thickness: subduction does not initiate. The maximum depth of yielding exceeds, however, the depth of the thermal lithosphere in the free-surface model and subduction initiates (Fig. 6
b). The same occurrence can be observed in the models applying a depth-dependent (Fig. 6
c, d) or a composite yield stress (Fig. 6
e, f). These experiments therefore show that subduction initiates once the model produces plastic failure down to the lithosphere-asthenosphere boundary.
The above results consider only small-scale convection, which is only one feature in a long list of possible triggers for subduction initiation in global models as the ones presented subsequently.
Subduction initiation by large-scale convection
Although global-scale models lack the spatial resolution of regional models, they do, in contrast, account for additional triggers of lithosphere-scale failure. Global-scale convective cells and narrow plumes rising from the core-mantle boundary (CMB) impinging the base of the lithosphere are just two of these mechanism that we focus on in the following sections.
Global stagnant-lid convection
In our global models, we apply both a stronger plate (than in the regional models) and a high plate-mantle viscosity contrast in order to prevent double-sided subduction and instead enable realistic, single-sided subduction (Crameri et al. 2012b). Necessitating a strong plate, single-sided subduction is therefore harder to initiate. Global-scale flow that produces wide mantle flow cells (i.e. >400 km) might, however, induce significantly higher lithospheric stresses and thus facilitate subduction initiation (e.g. Wong and Solomatov 2015). The global models presented in this section can, in contrast to the regional models above, where flow cells in the mantle are relatively narrow (e.g. <400 km), account for the effect of wider convection cells.
The purpose of our global 2D models is to reproduce the findings of our regional models regarding lithospheric deformation features. Figure 7 presents such an experiment with the free surface, while Additional file 1: Figure S1.7 presents the corresponding experiment with the free-slip surface. Here, we slightly increase the strength of the plate with a friction coefficient of μ=0.005 and a maximum yield stress of σ
y,const=21.7 MPa. We can indeed find the lithospheric criss-cross shear pattern again in the models where the surface evolves freely but not in the models where the surface is fixed vertically. Moreover, lithosphere-scale failure occurs early in the model evolution, even though these global models apply a slightly higher yield stress than the regional models. This supports previous findings (e.g. Wong and Solomatov 2015) that global mantle flow with larger flow cells adds a significant amount of additional forcing to the lid at the surface. The exact amount of the additional forcing has to be, however, carefully investigated in a future study.
Narrow hot mantle plumes
In general, models with realistic (e.g. based on laboratory measurements) friction coefficient and/or yield stress fail to break an intact stagnant lid. Even models that produce on-going single-sided subduction are, up to this point, not able to initiate subduction once in a stagnant-lid mode (e.g. Crameri et al. 2012b). The global 3D model presented in this section additionally includes narrow hot mantle plumes together with an even stronger plate that is, however, still relatively weak compared to laboratory estimates (see “Discussion” section for an extended discussion) using a friction coefficient of μ=0.09 and a maximum yield stress of σ
y,const=190.0 MPa. The experiment presented here with the free-slip surface (i.e. g
l
o
b
a
l3) therefore remains in stagnant-lid convection even after a long time of more than 6 Ga (bottom row of Figure 8 and Additional file 1: Figure S1.8). However, the corresponding experiment that additionally features a free surface (i.e. g
l
o
b
a
l4) spontaneously initiates subduction after starting in stagnant-lid mode (top row Figs. 8 and 9). An additional movie file shows this in more detail (see ). The mantle plumes are caused by the combined heating from the bottom (i.e. from the CMB) as well as from within. The isothermal CMB thereby favours the development of narrow plumes that are able to locally weaken the lithosphere and hence to facilitate subduction initiation as was previously suggested by Ueda et al. (2008) and Burov and Cloetingh (2010). This model additionally applies a depth-constant, ductile yield stress in combination with the depth-dependent yield stress (see Table 3). This helps to critically weaken the base of the plate and increases the impact that mantle plumes have on the cold plate.
Additional file 3: Movie of the experiment “global4” showing the temporal evolution of the global, fully spherical, 3D model with the hot temperature isosurface (red) and stiff viscosity isosurfaces (grey). (MOV 22192 kb)
Figure 9 illustrates the time evolution in multiple snapshots at the location of the initial subduction initiation inside the 3D spherical model. This event can be attributed to the following five phases. Phase 1: Local lithospheric thinning by plumes is characterised by narrow rising plumes impinging the base of the fully intact, stagnant lid. The additional heat at the lithosphere-asthenosphere boundary (LAB) from the hot upwellings causes the lithosphere to locally weaken and to thin out (Fig. 9
b). Phase 2: Large LAB topography develops by the continued local thinning caused by mantle plumes and the associated thickening nearby caused by the return flow of the hot upwelling. The return flow consists of a laterally limited horizontal component (connecting up- and downwelling) and a downward component. Both are causing significant topography in the base of the lithosphere (Fig. 9
c). Phase 3: An additional plume pulse by a reactivation of the upwelling causes a broad patch of hot material spreading at the base of the thinned lid. This further weakens the already thin plate and causes a decoupling of the thick part of the lithosphere from its thinned surroundings (Fig. 9
d). Phase 4: Plate failure is caused by the sinking of the thickened plate portion. The sinking of the plate can be related (i) to its negative buoyancy, (ii) to the adjacent, hot and failing part of the plate and (iii) to the support of the already existing downward return flow of the plume-driven convective cell (Fig. 9
e, f). Phase 5: Subduction finally takes place and the sinking slab rearranges the mantle flow in its surroundings: The typical poloidal flow in the mantle wedge and in the back-slab region and the toroidal flow around the slab edges form (Fig. 9
g).
The observation of these five phases indicates that the occurrence of LAB topography (phase 2) and the heating caused by hot mantle upwellings (phases 1 and 3) are here key to produce subduction initiation from a stagnant lid.
Global overturn event
Once a stagnant lid spontaneously initiates subduction, it can evolve either by an ongoing mobile lid or by an episodic lid that is characterised by temporally separated events with intermediate stagnant-lid phases (Tackley 2000a,b). In the latter case, a single episode of lithosphere recycling is termed a “global overturn”, which is believed to be the dominant mechanism in Venus’ dynamic evolution (e.g. Armann and Tackley 2012; Reese et al. 1999). Here, we present and discuss a global, fully spherical model that produces, for the first time, such a global overturn event.
The time evolution of the model with both internal and basal heating is presented in Fig. 10. This experiment starts from a globe-covering, intact lid that has no lateral heterogeneities (like plate thickness variations or slabs). Due to heating at the CMB, several point-like, hot upwellings (i.e. mantle plumes) form and cause locally round locations of weakening at the base of the lithosphere (t
E1). This weakening is enforced at later stages of the evolution and the former point-like plumes start to merge (hence reducing their total number) and deform to more elongated, sheet-like structures at time t
E2. The flow in the upper mantle is mainly dominated by these upwellings at this stage. The hot upwellings usually form thereby the centre of one of the several convection cells. At time t
E3, subduction has initiated at one spherical weak patch in the lithosphere that is close to but not exactly above one of the strongest hot anomalies. Single-sided subduction forms a horseshoe-shaped trench that kinks into one double-horseshoe-shaped structure and splits into a separately evolving subduction zone at time t
E4. The two subduction zones subsequently propagate throughout the model forming more complex and wider structures until a nearly intact belt of subduction is covering the globe (t
E5). Mantle flow is now dominated by the sinking portions of the plates. A few anomalously hot upwellings supported by the CMB still remain. Subduction stops again, shortly after the recycling of most of the former stagnant lid at time t
E6.
After spontaneously starting subduction from a stagnant lid, the model evolves with a mobile lid until the oldest and hence coldest and strongest portions (i.e. the former stagnant lid) are recycled into the mantle. Subduction acts quickly and so does not allow the newly created plate portions to cool down sufficiently to become sufficiently negative buoyant. These portions cannot therefore continue to subduct and the systems falls back into stagnant lid mode. This global overturn event operates around 1.2 Ga.
For such an event to happen, high plate velocities caused by a fast sinking of the slabs are needed. This can be enforced by having a small viscosity increase with depth (see Crameri et al. 2012b) or more generally by a lower viscosity mantle. The first is not the cause in this study, since other models presented here do not show a global overturn event although they have the same amount of viscosity increase with depth. The latter, a generally lower mantle viscosity, is, however, enforced by having a stagnant lid in the beginning. This is due to the fact that stagnant-lid convection is not as efficient at cooling the mantle than mobile-lid convection and thus heats up the mantle. We show that once in a stagnant-lid mode, a planet thus might preferentially evolve by temporally discrete global overturn events rather than by a continuous recycling of lid. This has strong relevance to Earth’s sister planet Venus, as it currently has no mobile lid, but is believed to recycle its surface episodically in such spectacular global-overturn events (e.g. Armann and Tackley 2012). The initial difference, however, that caused a stagnant-lid on Venus and a mobile lid on Earth might be explained by the absence of liquid water on Venus’ surface (Kaula 1999) and the impact of the hot, dense Venusian atmosphere, which can inhibit localisation of lithosphere-scale thermal shear faults (Crameri and Kaus 2010) or foster healing through increased grain-growth (Landuyt and Bercovici 2009).