The effect of rheological parameters on plate behaviour in a self-consistent model of mantle convection

Transcription

1 Physics of the Earth and Planetary Interiors 142 (2004) The effect of rheological parameters on plate behaviour in a self-consistent model of mantle convection Claudia Stein, Jörg Schmalzl, Ulrich Hansen Institut für Geophysik, Westfälische Wilhelms-Universität Münster, Münster 48149, Germany Received 26 February 2003; received in revised form 7 January 2004; accepted 7 January 2004 Abstract In a three-dimensional numerical model of mantle convection we have explored the self-consistent formation and evolution of plates. This was done by investigating the influence of temperature, stress and pressure dependence of the viscosity on plate-like behaviour. Further, the role of a depth-dependent thermal expansivity and of internal heating was examined. First-order self-consistent plate tectonics can be obtained for temperature- and stress-dependent viscosity: we observe virtually undeformed moving plates and the phenomenon of trench migration. However, plate motion only appears during short intervals, which are interrupted by long phases characterised by an immobile lid. An improvement is achieved by adding pressure dependence of the viscosity and of the thermal expansivity to the model. This leads to extended plates with continuous plate motion and the appearance of a low-viscosity zone. Internal heating accelerates subduction and gives an increase in plate velocity the plates are otherwise left intact. We find that the surface velocity is largely determined by subduction rather than by upwellings. Plate-like behaviour is restricted to a narrow parameter window. The window is bordered by an episodic or a stagnant lid type of flow on the one side and by a domain in which convection is akin to constant viscosity flow on the other side. With these different convective regimes the behaviour of various terrestrial planets can be described self-consistently. Occasionally even a change from the plate regime to the stagnant lid regime occurs. Such a type of transition has possibly changed the style of tectonics on some terrestrial planets, such as Mars Elsevier B.V. All rights reserved. Keywords: Self-consistent; Mantle convection; Plate tectonics; Rheology; Numerical modelling 1. Introduction Plate tectonics is the most prominent surface manifestation of the internal dynamics of the Earth. The particular type of surface behaviour which is found today seems to be a unique phenomenon among the terrestrial planets. While on Earth the lithosphere is split up into pieces moving relatively to each Corresponding author. Tel.: ; fax: address: stein@earth.uni-muenster.de (C. Stein). other, Mars is believed to be covered by a stagnant lid (Reese et al., 1998; Choblet and Sotin, 2001; Zuber, 2001) whereas plate tectonics might have existed in earlier times (Sleep, 1994; Nimmo and Stevenson, 2000). Similar circumstances are found on Mercury and the Moon, Venus on the other hand might have experienced global resurfacing events. In this scenario lithospheric plates collapsed into the mantle followed by the creation of new lithosphere due to conductive cooling (Turcotte, 1993; Strom et al., 1994; Fowler and O Brien, 1996; Reese et al., 1999a) /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.pepi

2 226 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) The connection to thermal convection in the planets mantle was established early, but the question which material properties would permit plate-like behaviour of the convective flow itself received substantial interest in the recent years. One essential point is the assumption of a strong temperature dependence of the viscosity because in such a way the formation of a rigid surface has already been achieved. Although laboratory experiments have been successful in the investigation of convection in fluids with temperature-dependent viscosity (Booker, 1976; Nataf and Richter, 1982; Richter et al., 1983; Davaille and Jaupart, 1993, 1994), they can hardly be devised to result in plate tectonics. Numerical experiments, becoming increasingly powerful over the last decade, were able to provide insight into many mantle-relevant configurations, among them extremely strong temperature dependence of the viscosity (Torrance and Turcotte, 1971; Christensen, 1984b; Ogawa et al., 1991; Hansen and Yuen, 1993; Tackley, 1993, 1996; Moresi and Solomatov, 1995; Solomatov and Moresi, 1996, 2000; Trompert and Hansen, 1998a). The presentation of a model, however, in which plates evolve self-consistently as a result of mantle convection, turned out to be a formidable problem. Facing the difficulties which were associated with the self-consistent simulation of plates, many authors decided to impose plate-like features in an ad hoc fashion. An obvious way is to prescribe plate velocities by the means of specific boundary conditions (Lux et al., 1979; Olson and Corcos, 1980; Hager and O Connell, 1981; Davies, 1988; Lithgow-Bertelloni and Richards, 1995). These are the so-called kinematic approaches. Later, in the dynamical approaches the plate motion was not prescribed, but obtained dynamically by the distribution of buoyancy in the mantle (Schmeling and Jacoby, 1981; Gurnis, 1988; Davies, 1989; Ricard and Vigny, 1989; King and Hager, 1990; Gable et al., 1991; Zhong and Gurnis, 1996; Zhong et al., 2000; Monnereau and Quéré, 2001). These models either impose rigidity of the plates and/or introduce weak zones. In a further step dynamically evolving boundaries were obtained (Zhong and Gurnis, 1995a,b; Lowman and Jarvis, 1999). Both these approaches serve to investigate the influence of plates on several aspects of mantle dynamics, for example, on the destruction of chemical reservoirs in the mantle (Ferrachat and Ricard, 2001) or the thermal structure beneath a moving plate (King et al., 2002). In a different approach no geometrical constraints are put onto the surface but a rheological model is employed which comprises a physical formulation of the dependence of various parameters on external forces. The main goal of these self-consistent models is to view plates and the mantle as being one integrated system. This approach today still faces severe limitations due to the available computational power, they nevertheless give a first understanding of the way plate-like features are generated. Besides the dependence of the viscosity on temperature, a dependence on stress has been especially considered. If the stresses applied exceed the mechanical strength of the lithosphere, deformation appears (Fowler, 1993). The governing laws for the stress dependence of the viscosity, however, are not fully understood, so that a variety of approaches exist. Non-Newtonian rheology was explored by Cseperes (1982), Christensen (1983, 1984a), Christensen and Harder (1991), Weinstein and Olson (1992), Bercovici (1993, 1996) and Solomatov and Moresi (1997). For the investigation of Newtonian behaviour, deformation mechanisms such as strain-weakening, selflubrication and viscoplastic-yielding were employed (Bercovici, 1996, 1998; Moresi and Solomatov, 1998; Trompert and Hansen, 1998b; Tackley, 2000b,c). By using a strongly temperature-dependent viscosity together with an appropriate yield stress, Trompert and Hansen (1998b) were able to obtain self-consistent plate-like behaviour in a fully three-dimensional flow. In their model, however, plate-like motion only appeared episodically, interrupted by long periods during which a stagnant lid existed. Similar behaviour was observed by Tackley (2000b). Perhaps this timedependent mode of plate tectonics has at one time existed on Earth today we observe a more continuous motion of the plates. In the meantime, there are a few models which allow for the self-consistent simulation of plate tectonics (Tackley, 2000b,c) but it is clear that these approaches are still in their infancy (Tackley, 2000a). As mentioned above this type of model is extremely demanding with respect to computational power: each presented run with a resolution of took about 6 8 months on a Pentium IV workstation. Tackley (2000b) highlights the dilemma: either a small number of state of the art cases can be run or a large number of more routine cases. Still there can be

3 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) no question that self-consistent models are necessary in order to understand the integrated system formed by the mantle and the lithosphere. In this paper we explore the evolution of the system for different rheologies with particular respect to plate-like behaviour. Our main goal is to identify conditions under which plate-like surface motion exists. To do so, we start out with a careful analysis of temperature- and stress-dependent rheology, where the stress dependence is applied in form of viscoplastic yielding. In a further step we consider a viscosity that increases and a thermal expansion coefficient that decreases throughout the model domain, because laboratory experiments clearly indicate that the viscosity and the thermal expansion coefficient are also functions of pressure (Kirby, 1983; Chopelas and Boehler, 1989; Mackwell, 1991; Chopelas and Boehler, 1992; Karato and Wu, 1993). Earlier studies (Gurnis and Davies, 1986; Hansen et al., 1991, 1993; Leitch et al., 1991; Balachandar et al., 1992; Bunge et al., 1996, 1997; Dumoulin et al., 1999; Lowman et al., 2001; Richards et al., 2001) have revealed the strong influence of these variations on the planform of convection which can have a potentially profound impact on the surface motion. Tackley (2000c) has already documented that the pressure dependence of the viscosity certainly changes the style of plate motion. We here combine the pressure dependence with a decreasing expansion coefficient and demonstrate how continuous even steadily moving plates can be obtained. Finally, the relative importance of internal heating (provided by radioactive elements in the mantle) to bottom heating (provided by heat loss from the core) is known to strongly influence the style of convection (Turcotte and Schubert, 1982). Most numerical studies, taking into account the effect of internal heating on plate-like behaviour, employed adiabatic conditions at the lower boundary (Houseman, 1988; Bercovici et al., 1989; Parmentier et al., 1994; Grasset and Parmentier, 1998; Tackley, 2000b,c), i.e. the heat flux at the core mantle boundary was assumed to vanish. In this paper, we adopt a more realistic scenario similar to Weinstein et al. (1989), Travis et al. (1990), Weinstein and Olson (1990), Hansen et al. (1993), Sotin and Labrosse (1999) and Lowman et al. (2001) in which internal heating is taken into consideration while at the same time a constant temperature at the core mantle boundary is assumed. We start in Section 2 by describing the model equations, the numerical implementation and the model geometry. This is followed by the description of the rheological model in Section 3 and the definition of some values which are relevant for determining Earth-like behaviour in Section 4. The results of the various assumptions are presented and discussed in Section 5. Finally, we give a summary in Section Mathematical model We consider thermally driven convection in a Boussinesq fluid with infinite Prandtl number in a three-dimensional Cartesian geometry. The equations describing this problem can be written in the following non-dimensional form: T + u T 2 T = Q, (1) t u = 0, (2) p + σ + RaTe z = 0 (3) with T the temperature, t the time, u = (u, v, w) T the velocity vector with the components u, v and w being the velocities in the coordinate directions x, y and z, respectively. p is the pressure without the hydrostatic component and σ is the deviatoric stress tensor: ( ui σ ij = 2ηė ij = η + u ) j (4) x j x i with η the dynamic viscosity and ė ij the strain-rate tensor. e z is the unit vector in z-direction and Q is the rate of internal heating: Q = Ra H Ra = Hd2 k T (5) with H the rate of internal heat generation per unit volume, d the thickness of the convecting layer, k the thermal conductivity and T the temperature difference across the convecting layer. Ra H is the Rayleigh number due to internal heating and Ra the Rayleigh number based on the temperature drop between upper and lower boundary. The definition of the Rayleigh number for varying parameters is not unique. One can either use a Rayleigh number based on the top or bottom parameters or a Rayleigh number based on the interior values.

4 228 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Moresi and Solomatov (1995) suggest that a Rayleigh number based on the interior viscosity characterises convection better. However, the interior Rayleigh number is not very convenient as it cannot be prescribed a priori. In our model the system is therefore characterised by a surface Rayleigh number: Ra = α 0gρ 0 Td 3, (6) η 0 κ which is defined by using the thermal expansivity α 0, the viscosity η 0 and the density ρ 0 at T = 0. g is the acceleration due to gravity and κ the thermal diffusivity. Since the numerical method is fully described in Trompert and Hansen (1996), we only give a brief summary here. The equations are spatially discretised by using a control volume method on a staggered grid described by Harlow and Welch (1965). Grid spacing is uniform in horizontal directions. A Chebyshev grid is employed in vertical direction to approximately resolve the thin thermal boundary layers at high Rayleigh numbers. The fully implicit Crank Nicolson method is used to integrate the energy equation (Eq. 1) in time. Finally, the resulting system of nonlinear equations is solved every time-step with the multigrid method. As smoother we use the SIMPLER method by Patankar (1980) and W-cycles to visit the coarser grids. Numerical experiments with a strongly variable viscosity can lead to multigrid convergence problems (Tackley, 2000b). By solving the pressure and pressure-correction more precisely, in our model carried out in the form of a multigrid iteration with the Jacobi method as smoother instead of simple Jacobi iteration, this problem can be avoided. As iteration method for the temperature we use Jacobi and for the velocity Gauss Seidel. In this study a three-dimensional domain with an aspect ratio (width and depth over height) 2 was used with a resolution of Comparisons with single experiments performed on a grid revealed no qualitative differences. These runs were taken from the stagnant lid regime with a combined temperature- and stress-dependent viscosity, from the episodic regime with brittle deformation and from the plate regime with additional pressure-dependent viscosity in order to check if the general behaviour is reflected. The boundaries were stress-free and isothermal on top and bottom, and reflecting conditions have been employed at the sides. All the calculations presented in this work were started with an initial temperature of This resembles the interior temperature found in experiments with purely thermoviscous convection in the stagnant lid regime if the same viscosity contrast and Rayleigh number as used in this work is applied. With this initial temperature the formation of the stagnant lid developed somewhat faster than if the conductive temperature profile was used as an initial condition. In general our preliminary runs with different initial temperatures and form of added perturbation resulted in the same model behaviour. Only at the transitions between different regimes the system sometimes preferred one regime to the other. 3. Rheological model Experiments on rock deformation have shown that the viscosity of the Earth s and other planets mantle is a function of different parameters such as temperature, stress and pressure (Karato and Wu, 1993), whereupon temperature dependence has the strongest effect on viscosity and is usually expressed by the Arrhenius law: [ ] E η(t) = C exp (7) RT with T the temperature, C a constant, E an activation energy and R the gas constant. Frank-Kamenetskii (1969) showed that the law can be simplified by using an approximated exponential function: η T = exp( rt), (8) where r = ln(η) and η = η 0 /η T being the viscosity contrast over the domain, with η 0 = 1 the viscosity at the surface (for T = 0) and η T at the bottom (for T = T ). This form of temperature dependence is commonly used (e.g. Ogawa et al., 1991; Ratcliff et al., 1995, 1997; Grasset and Parmentier, 1998; Moresi and Solomatov, 1998) and also employed in this study. The stiff surface, as resulting from temperaturedependent viscosity, can be deformed by further assuming a viscoplastic rheology. In this case the material is rigid for stresses lower than the yield stress

5 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) and for stresses larger than the yield stress failure occurs. If the material fails by fracture, the deformation is termed brittle and in case of plastic flow, it is termed ductile (Ranalli, 1987). Both brittle and ductile deformation occur in the lithosphere, because various deformation mechanisms are dominant at different depths. In shallow depths, rocks are brittle, where the strength is proportional to depth (Byerlee, 1968). In the middle a ductile mechanism is predominant, where strength is almost constant with depth (Kirby, 1983) and, finally, in the deeper lithosphere a power-law flow is the dominant mechanism of failure (Kohlstedt et al., 1995). As already pointed out in Section 1, there are various approaches for the inclusion of a stress dependence. We make use of a Bingham model that is based on the one used in Burgess and Wilson (1996) for which the effective viscosity is given as: 2 η = (9) 1/η T + 1/η σ with the temperature-dependent viscosity η T specified in Eq. (8) and a stress-dependent part η σ : η σ = η + σ Y(z). (10) (ɛ : ɛ) 1/2 σ Y (z) is the yield stress, (ɛ : ɛ) 1/2 is the second invariant of the strain-rate tensor and η (plastic viscosity) is the effective viscosity at higher stresses. In this research the plastic viscosity is chosen as η = 10 5,so that in this way the surface is stagnant but can readily be deformed when the yield stress is chosen correctly, i.e. it is so low that the stresses in the material can easily overcome the yield stress. The precise critical value, however, is dependent on various parameters such as viscosity contrast or Rayleigh number. This dependence on parameters will be discussed below, e.g. Fig. 2 displays that the critical yield stress, at which deformation occurs, decreases with increasing viscosity contrast. The yield stress σ Y (z) represents the value at which deformation of the surface occurs and is defined by Byerlee s law (Byerlee, 1968): σ Y (z) = σ 0 + σ z (1 z), (11) consisting of the yield stress at the surface σ 0 and a depth-dependent part σ z. Here z is the depth of the domain with z = 1 at the surface and z = 0 at the bottom. The deformation is considered to be ductile if the yield stress is constant, and brittle for a depth-dependent yield stress. Thus in case of σ z = 0 the constant yield stress represents ductile deformation, otherwise brittle deformation is used. Both cases have been considered in this study. Furthermore, the viscosity of mantle material is affected by pressure (Kirby, 1983; Mackwell, 1991; Karato and Wu, 1993). Therefore a pressuredependent viscosity was added to the model. The viscosity in this case is commonly given by changing the Arrhenius law to: [ ] E + pv η(p, T) = C exp. (12) RT Here, p is the pressure and V the activation volume. With respect to the Boussinesq approximation (i.e. constant density) the pressure dependence is converted to a depth dependence, so that the Frank-Kamenetskii approximation (Eq. (8)) has to be modified in the following manner: η p,t = exp[ rt + c(1 z)]. (13) c = ln(rd) and RD is the measure of the pressure (i.e. depth) dependence. Further experiments showed that not only the viscosity varies over the mantle but also other parameters such as the thermal expansivity. First investigations on the variation of thermal expansivity with pressure for mantle silicates were done by Birch (1968) and a substantial decrease in the thermal expansivity across the mantle by a factor of about 5 10 (Chopelas and Boehler, 1989, 1992) was found. We have therefore also assumed a depth-dependent thermal expansivity α in the form of: ( α(z) = α 0 m(1 z) + 1 ) 3 (14) with α 0 the expansivity at the surface and m the factor of depth dependence. Depth-dependent thermal expansivity has already been explored (Zhao and Yuen, 1987; Hansen et al., 1991, 1993; Balachandar et al., 1992), but its effect on plates was not considered in these models. 4. Diagnostic values In order to provide a qualitative description of plate-like behaviour we employed various character-

6 230 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) istical diagnostic values, some of which have been chosen following Tackley (2000b,c) for a convenient comparison. Regarding Earth, plate-like motion means that the surface should move with a uniform velocity and with only little internal deformation, i.e. a rigidly moving surface should result. Deformation only occurs in narrow surrounding zones. Therefore it seems useful to calculate the amount of surface deformation and the surface velocity Surface velocity v surf The surface velocity is calculated as the root mean square of the horizontal velocity components (the vertical component is comparatively small and can be neglected) at the surface as: v surf := u 2 + v 2. (15) To enable the comparison with the Earth, we calculate the surface velocity with the scaling-factor κ/d 10 3 cm/yr. κ = 10 6 m 2 /s is the thermal diffusivity and d the height of the domain which is, for the whole mantle convection, d = 2900 km Surface deformation ɛ surf The surface deformation is determined by the square root of the second invariant of the strain-rate tensor at the surface given by: [ ( u ) 2 ɛ surf := + x ( ) v ( u y 2 y + v ) ] 2 1/2. x (16) According to Tackley (2000b), regions displaying less or equal ɛ plate := 0.2ɛ max surf are regarded as plates while regions with more than 20% of the maximum surface deformation are considered as plate boundaries. If ɛ plate is very low, we thus have a rigid surface (i.e. plate) but for a high ɛ plate the surface is strongly deformed, i.e. fluid-like Plate-criterion Besides exhibiting little deformation, plates should also move with an almost uniform velocity. Therefore, we do not only use the surface deformation but also the horizontal divergence: h u = u x + v y, (17) as an important diagnostic measure. The horizontal divergence should be close to zero within plate regions, while a positive divergence characterises upwellings and a negative divergence appears at convergent plate boundaries. A small amount of deformation within a plate and small variations of the velocity in the plate region are clearly tolerable. Therefore a small deviation of the horizontal divergence from zero is also tolerable, so that the region exhibiting less than 20% of the maximum absolute divergence is regarded as plate. Thus to summarise our plate-criterion: we consider a region to be a boundary (P b ) when more than 20% of the maximum deformation (ɛ >ɛ plate ) and more than 20% of the maximum absolute divergence is reached. The rest is made up by either the stagnant lid (P s ), if surface velocities are low, or by plates (P p ) when 80% of the maximum surface velocity (v plate = 0.8v max surf )is reached Boundary plate ratio R p According to the plate-criterion we can determine the ratio of boundaries to plates, which is the boundary-size P b divided by the plate-size P p. Gordon and Stein (1992) found that the surface area of the Earth with little internal deformation and uniform velocity takes up 85%, while 15% is given as wide boundaries. Thus a realistic boundary plate ratio is R p Toroidal poloidal ratio R TP In addition to the boundary plate ratio, it is also interesting to know of what type the boundaries are. The divergent and convergent boundaries are associated with a poloidal flow, whereas the transform-faults are associated with a toroidal flow. Convection with vanishing mechanical inertia is a purely poloidal flow, however, toroidal motion can be excited by lateral variations of the viscosity (Bercovici et al., 2000). The ratio of toroidal to poloidal energy has been subject of many investigations (e.g. Gable et al., 1991;

7 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Olson and Bercovici, 1991; Ribe, 1992; Lithgow- Bertelloni et al., 1993; Balachandar et al., 1995; Bercovici, 1995; Weinstein, 1998) and is believed to vary between 0.25 and 0.5 over the last 120 million years (Lithgow-Bertelloni et al., 1993). In numerical models the ratio can be calculated from the two-dimensional horizontal velocity field: u h = u h,pol + u h,tor = h Φ + h (Ψe z ). (18) This splitting is equivalent to the Helmholtz decomposition in a curl-free divergence field and in a solenoidal rotational field. The poloidal and toroidal parts can then be calculated by computing the horizontal divergence and the vertical vorticity of Eq. (18), respectively. As the rotational field (toroidal part) has to be solenoidal, it can be neglected in the calculation of the horizontal divergence. This leads to a Poisson equation, from which the poloidal part can be computed: h 2 Φ = h u. (19) Since the divergence field is curl-free the poloidal part is irrelevant for the determination of the vertical vorticity: h ( h Ψe z ) = h 2 Ψe z = ( u) h. (20) The ratio of toroidal poloidal energy in the surface is finally given as: ( h Ψ) R TP = 2 ( h Φ) 2 (21) with Ψ the toroidal and Φ the poloidal component Plume velocity v u or v d In our experiments the velocities of up- and downwellings vary strongly for different rheologies. In order to qualify the strength of the up- and downwellings better, we determine the vertical velocities of each run at a height of z = 0.5 (mid-depth). The velocity of the upwelling v u is determined as the maximum velocity in this plane and the velocity of the downwelling v d as the minimum velocity. 5. Results In this section we will discuss our results starting from a simple rheology and step by step pass over to a more complex one. After summarising the main findings of previous studies on thermoviscous convection, we will discuss our results obtained with a combined temperature- and stress-dependent rheology. First a constant yield criterion representing ductile deformation is considered, followed by a depth-dependent yield stress representing brittle deformation. Additional depth dependence will then be discussed, namely first a depth dependence of the viscosity and secondly a depth-dependent thermal expansivity. Furthermore, we will investigate the influence of internal heating on the organisation of plates. In all cases we found three different convective regimes when varying the yield stress: a stagnant lid regime, a mobile lid regime and an episodic regime. For the sake of completeness, we will once present an example of all three regimes. For the subsequently applied rheologies we concentrate on the regime exhibiting the most plate-like behaviour. The model parameters of all runs are summarised in Table 1, which also provides an overview of the diagnostic values. A good plate-like behaviour is obtained when the plate velocity v plate is about a few cm per year and the ratio of toroidal to poloidal energy R TP is between 0.25 and 0.5 (Lithgow-Bertelloni et al., 1993). The values to be expected for the deformation of the plate ɛ plate, the boundary-size P b and the plate-size P p are illustrated by some case studies. For isoviscous convection (i.e. a fluid-like surface) with a Rayleigh number of Ra = 10 7 we find a plate deformation of ɛ plate = For thermoviscous convection with a viscosity contrast of η = 10 5 and a top Rayleigh number of 100 (i.e. a bottom Rayleigh number of 10 7 ) the surface is highly viscous and a rigid lid covers the complete surface. In this case the deformation of the plate is ɛ plate = 0.8. Thus we would expect an ideal plate, i.e. a rigid surface piece, to have a deformation similar to that observed for the rigid lid convection. However, the plate deformation ɛ plate for single plates surrounded by plate boundaries will be slightly higher, as little internal deformation is found in plates because of the stresses transmitted from the plate boundaries through a plate. In order to demonstrate the interpretative power of the boundary-size and plate-size, we constructed a hypothetical surface velocity field in the form of a boxcar type function for

8 232 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Table 1 Input parameters and diagnostic values, over bar values denote time-averaged values and the others the ones at one point of time Parameters Diagnostic values σ 0 σ z RD α Q v plate ɛ plate P b P p R TP v u v d Run Run Run Run Run Run Run / two plates: f(x, y) = 4 π m n=1 sin([2n 1]πx), 0 x, y 2. 2n 1 (22) By truncating the Fourier series representation after a specific number of terms m, the surface velocity evolves from a sine function (m = 1) to two uniformly moving plates with a sharp boundary (m = 500). Fig. 1 illustrates the surface velocity profile for these two extreme cases. In case of the sine function shaped surface velocity field we find a boundary-size of P b = 0.84 and a plate-size of P p = It is understandable that for this kind of surface velocity field plates are still found, because our plate criterion allows slight variations in the deformation and velocity of a plate. Truncating the Fourier series approximation after the second term (m = 2) reduces the boundary-size to P b = 0.66 and increases the Fig. 1. Synthetic velocity profiles for a Taylor series expansion of a boxcar function truncated after different terms. For m = 1a sine function results, for m = 500 the boxcar function represents two rigidly moving plates with an infinitely narrow boundary. plate-size to P p = For m = 10 we find P b = 0.08 and P p = For m = 500 two ideal plates (P p = 0.97) with a strongly localised boundary (P b = 0.03) are observed Temperature-dependent viscosity A rheology with a strong temperature dependence of the viscosity forms the basis of the rheologies applied in this study. Therefore different aspects and implications of the temperature-dependent viscosity explored in several other studies will be briefly presented in this section. Different approaches were used to examine the effect of convection with temperature-dependent viscosity, which will henceforth be referred to as thermoviscous convection. Booker (1976), Nataf and Richter (1982), Richter et al. (1983) as well as Davaille and Jaupart (1993, 1994) performed laboratory experiments, whereas a boundary layer theory was put forward by Morris and Canright (1984) and Fowler (1985). Numerical experiments are an alternative method that, compared to laboratory experiments, rather easily allow a change in the parameters and domain size. Some of these experiments were done in a two-dimensional (2D) Cartesian geometry (Torrance and Turcotte, 1971; Christensen, 1984b; Hansen and Yuen, 1993; Moresi and Solomatov, 1995; Solomatov and Moresi, 1996, 2000), others in a three-dimensional (3D) Cartesian geometry (Ogawa et al., 1991; Tackley, 1993, 1996; Trompert and Hansen, 1998a) and some in a spherical geometry (Ratcliff et al., 1996, 1997; Reese et al., 1999b). The strength of temperature dependence is given by a viscosity contrast between upper and lower bound-

9 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) ary and with increasing viscosity contrast the system runs through three different regimes (Solomatov, 1995): the low-viscosity contrast regime, the transitional regime and the high-viscosity contrast regime. For a low-viscosity contrast ( η < 10 2 ) thermoviscous convection resembles isoviscous convection, so that no plate-like behaviour is found. In the transitional regime (10 2 < η<10 4 ) a Rayleigh number dependent boundary divides the other two end-members: (1) the low-viscosity contrast regime for high Rayleigh numbers and (2) the high-viscosity contrast regime for low Rayleigh numbers (Hansen and Yuen, 1993). The high-viscosity contrast regime ( η > 10 4 ) results in a rigid upper layer. Here, negative buoyancy forces in the surface cannot overcome the viscous forces, so that the layer does not take part in the convective cycle. Furthermore the rigid layer becomes immobile, which led to the name stagnant lid convection. Conduction becomes the main heat transport mechanism within the upper layer. Beneath the stagnant lid the convection takes place in a virtually isoviscous medium. The inner temperature is comparatively high, since the stagnant lid shields the interior against effective cooling. To conclude, thermoviscous convection with high viscosity contrasts can be considered to be a first step in the self-consistent incorporation of plates into convection models. The surface no longer behaves fluid-like but becomes rigid. However, single plates have not been observed Stress-dependent viscosity: (a) ductile deformation The rigid surface formed as a result of a viscosity contrast of η = 10 5 will now be deformed and mobilised by applying a stress-dependent viscosity. In this work a viscoplastic yielding with a constant and depth-dependent yield stress is considered. At first the effect of a constant yield criterion on a fully developed stagnant lid is investigated. Varying the yield stress we obtain three regimes, namely, the mobile lid regime for low yield stresses, a regime with episodic behaviour for moderate values and the stagnant lid regime for high yield stresses. Similar results achieved in a two-dimensional domain have been reported by Moresi and Solomatov (1998). Tackley (2000b) also reported the finding of these regimes in a three-dimensional study, however, for a different type Fig. 2. Different regimes of convection as function of (a) the viscosity (b) and surface Rayleigh number. Transitions from the mobile lid regime to the episodic regime occur at σ me and from the episodic regime to the stagnant lid regime at σ es. For parameters used see Appendix A. of stress dependence. The existence of these regimes seems thus fairly typical. Before discussing each regime in detail we first pay closer attention to the transitions between the regimes. In order to describe the transitions for different viscosity contrasts more precisely, we mapped out the space, that is spanned by the yield stress and the applied viscosity contrast (Fig. 2a) in a series of numerical experiments. This required the investigation of a large number of runs. A detailed overview of the runs conducted is provided in the first table of Appendix A. The transition from the mobile lid regime to the episodic regime occurs at a yield stress σ me and the transition from the episodic to the stagnant lid regime at σ es. As expected, for a low viscosity contrast the mobile

10 234 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) lid regime dominates, while for a high viscosity contrast the stagnant lid regime prevails. The reason is that at a sufficiently high viscosity contrast the stress dependence can hardly break the lid, resulting in the stagnant lid scenario. The effect of the Rayleigh number variation on the regime transitions is summarised in Fig. 2b. For this investigation a fixed viscosity contrast of η = 10 5 was assumed. Further details on the input parameters are listed in the second table of Appendix A. Increasing the Rayleigh number at a given yield stress leads to a transition from the stagnant lid to the mobile lid regime. This result is akin to the findings of Hansen and Yuen (1993) for thermoviscous convection. They observed that a stagnant lid exists for low Rayleigh numbers but that the flow changes to virtually isoviscous convection at sufficiently high Rayleigh numbers in the sense that the lid is mobilised and the internal temperature drops to 0.5. In this investigation with a stress-dependent rheology, both regimes are being separated by a region in which episodic behaviour occurs. Apart from the transitions between the regimes due to a change in parameters, we further observed a change from one regime to another within time. A yield stress which is close to the critical value σ es,at which a transition from the episodic to the stagnant lid regime occurs, leads to an evolution characterised by a transient behaviour. Initially episodic behaviour evolves before finally falling into the stagnant lid mode of convection. Such a change indicates, that it is dynamically plausible that a planet can change during its lifetime from one regime to another. A potential Fig. 3. Temperature field from the stagnant lid scenario (run1). Temperatures are colour-coded between the dimensionless values of zero (blue) and one (red). candidate for such a change could be Mars which is believed to be in the stagnant lid regime today but may have shown plate tectonics earlier in its evolution (Sleep, 1994; Nimmo and Stevenson, 2000). The single regimes will now be discussed separately on the basis of one example each Run1 (stagnant lid regime) Fig. 3 shows a snapshot of the temperature field as obtained with a high yield stress (parameters: η = 10 5, Ra = 100, σ 0 = 10 and σ z = 0). As the cold upper layer no longer takes part in the convective circulation and thus shields the interior against cooling, we find a fairly hot interior. The resulting internal tem- Fig. 4. Snapshot as obtained in the mobile lid regime (run2). (a) Temperature field and (b) surface motion. The white arrows represent the surface velocity.

11 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) perature is 0.92 which resembles the values found by Stengel et al. (1982) or Trompert and Hansen (1998a) for thermoviscous convection with the same viscosity contrast and Rayleigh number. Applying the plate diagnostics, we find that the surface velocity as well as the surface deformation are virtually zero (see Table 1). Furthermore the ratio of toroidal to poloidal energy is low (R TP = ). Expectedly, the plate-criterion gives a value of P s = 1 which means that 100% of the surface is covered by the stagnant lid. Since the central topic of this study is the appearance of self-consistent plate-like surface motion and plate boundaries we discard the stagnant lid regime in the subsequent sections of the paper Run2 (mobile lid regime) Reducing the value of the yield stress from σ 0 = 10 to σ 0 = 0.1 leads to a fundamentally different behaviour. The previously existing stagnant lid is now mobilised since the stress dependence can overcome the effects of the temperature dependence. In Fig. 4a a snapshot of the temperature field for the parameter combination of η = 10 5, Ra = 100 and σ 0 = 0.1 is displayed. Differently from the previous case, we find single up- and downwellings, where the upwellings even thrust through the uppermost layer. The surface now takes part in the flow and thus the interior is significantly cooler (T int = 0.5 instead of 0.92). This regime is virtually identical to isoviscous flow. The white vector arrows in Fig. 4b display the surface velocity, indicating that only small pieces of the surface move with almost the same velocity in the same direction, i.e. extended plates do not exist. A closer look at the diagnostic values (see Table 1) reveals that in this regime plate boundaries have formed (P b = 0.18, i.e. 18% of the surface is made up by plate boundaries). The rest of the surface consists of small pieces which are strongly internally deformed (ɛ plate = ). This value is similar to the deformation found for isoviscous convection with the same bottom Rayleigh number (ɛ iso plate = ). Thus the surface behaves rather like a fluid than as plates. The surface moves relatively fast (v max = 1.78 cm/yr). The ratio of toroidal to poloidal energy (R TP = ) is slightly increased because the surface behaves fluid-like and thus shear motion can appear. To summarise, we find that for low values of the yield stress, a combined temperature- and stressdependent rheology yields the so-called mobile lid regime. This is, however, characterised by a surface behaving fluid-like rather than plate-like and resembles isoviscous convection Run3 (episodic regime) Between these two regimes we find a parameter region which is characterised by an episodic behaviour. In this regime, periods in which a stagnant lid exists are interrupted by phases during which the surface is mobilised through the stress dependence. This phenomenon has, in principle, already been reported by Trompert and Hansen (1998b), but we will here further quantify this regime by applying the plate diagnostics. Fig. 5 displays several snapshots of the temperature field from an experiment within this regime. The parameters of this run were: η = 10 5, Ra = 100, σ 0 = 2 and σ z = 0. The yield stress σ 0 was chosen to be higher than that for the mobile lid regime but lower than the critical value σ es above which a stagnant lid forms. The first snapshot (Fig. 5a) was taken when the stagnant lid has fully developed. While the stagnant lid thickens, the stresses increase and stress dependence becomes stronger than temperature dependence. Subsequently, this leads to failure of the stagnant lid and results in the breaking open of the surface. Figs. 5b h show the evolution of a cold descending current at the front plane and of a hot ascending current at the front edge. When the sinking current spreads out at the bottom, the surface location of the downwelling moves from the middle to the left. This feature is similar to trench migration a phenomenon typically observed on Earth. Finally, the downwelling tears off (Fig. 5i) and a new lid forms by conduction (Fig. 5j). Between the up- and downwelling the surface starts to move and what is important, it moves like a plate (indicated by means of the grey surface in Figs. 5b h). This, in particular, means that this part of the surface moves with a uniform velocity and that it further shows only little internal deformation. Fig. 6a and 6b present the surface velocity and the surface deformation for one time instant (corresponding to Fig. 5c). The plate-like behaviour is clearly revealed: a part of the surface moves with constant velocity (Fig. 6a) and the velocity changes abruptly at the edges of that part. Furthermore, the moving part is almost undeformed

12 236 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Fig. 5. Temperature distribution for one complete overturn for a run with ductile deformation (run3). The grey area illustrates the uniformly moving surface.

13 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Fig. 6. Diagnostic values for one run in the episodic regime: (a) distribution of the surface velocity and (b) distribution of the surface deformation at the time of the snapshot in Fig. 5c. (c) Time history of the surface velocity and (d) time history of the boundary-size. (Fig. 6b), all deformation takes place at the boundaries. The part of the surface mimicking plate-like behaviour is, however, small. At the given time instant the plate covers only 9% of the surface and further 6% are plate boundaries. Most of the surface is still covered by a stagnant lid. The time-averaged values of the plate-size P p and the boundary-size P b,given in Table 1, indicate that boundaries are more strongly localised than in the mobile lid regime. The plate-size is smaller than in the mobile lid regime, but we will subsequently show that the plates are more rigid. The episodic character of this phenomenon may be the most unrealistic feature. While it seems clear, that on Earth plate tectonics is a time-dependent process, at least todays plate tectonics does not show such a strong episodic behaviour as observed in the model. Fig. 6c shows a time history plot of the surface velocity. Obviously, long phases of stagnant lid style are interrupted by a few sharp bursts, during which a significant surface velocity appears. Similarly, plate boundaries (Fig. 6d) only appear during these short periods. The time-averaged values of the plate velocity and surface deformation are given in Table 1. When scaled to Earth values we obtain a plate velocity of v plate = 1.3 cm/yr which lies within realistic limits. Furthermore, the plates are less internally deformed ( ɛ plate = ) than they were in the mobile lid regime ( ɛ plate = ). Thus this regime yields rigidly moving plates. The episodic character is also revealed by the time history plot of the toroidal poloidal energy (Fig. 7). A toroidal component is only present during the phase in which the plate moves. Then the plate slides along R TP 0.08 (d) (e) (f) (c) (b)(g) 0.04 (h) 0 0 (i) (a) (j) time Fig. 7. Ratio of toroidal poloidal energy R TP as a function of time. The dots mark time instants displayed in Fig. 5.

14 238 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) V z x e+03 1e Fig. 8. Distribution of the vertical velocity at depth z = 0.5 for the case shown in Fig. 5c. the remaining stagnant lid a feature which resembles strike-slip motion. A maximum value of R TP = 0.17 is reached which is lower than the value of 0.25 proposed by Lithgow-Bertelloni et al. (1993) for the Earth, but we still find that toroidal motion does appear in convection models if lateral variation in the viscosity is considered. In order to explain the episodic behaviour, we tried to quantify the relative importance of the up- and downwellings. This was done by examining the distribution of the vertical velocity in the plane z = 0.5, i.e. at mid-depth. In Fig. 8 the distribution is portrayed, again at the time instant given in Fig. 5c. We find a maximum positive vertical velocity (v u = ) y where the upwelling breaks through the surface and the highest negative value (v d = ) in the downwelling. Besides these two significant places, we find some minor variations in the vertical velocity. Compared with our other results these values are relatively high. However, the fast subduction seems to be the reason for fairly fast moving plates. We conclude this from a further case (not shown here) where a similar upflow velocity (v u = ) but reduced downflow velocity (v d = 49.52) resulted in a likewise reduced surface velocity ( v plate = 0.08 cm/yr). In that case a more steady plate behaviour was reached, while in the example shown here the plate-like area is destroyed much faster by subduction than created by the upwelling. Finally, for the temperature- and stress-dependent rheology Fig. 9 summarises the behaviour of the plate diagnostics in dependence on the yield stress. Presented are the time-averaged values of the surface velocity (solid line), the ratio of boundary to plate (dotted line) and the ratio of toroidal to poloidal kinetic energy (dashed line). The figure clearly displays the three regimes: all three values increase slightly in the mobile lid regime (σ <2) and reach a maximum in the episodic regime (σ = 3 4). A clear change is found at the transition to the stagnant lid regime because then the immobile lid (v surf small) covers the whole surface (P s = 1, i.e. R p = 0). As neither strike-slip motion nor rotation appears diagnostic value o o Vsurf Rp R TP yield stress Fig. 9. Diagnostic values (surface velocity v surf, boundary plate ratio R p and ratio of toroidal poloidal kinetic energy R TP ) in dependence on the yield stress. The input parameters of the runs were: η = 10 5 and Ra =

15 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) in the surface, the toroidal poloidal ratio R TP is small. Thus, a combined temperature- and stress-dependent rheology produces the most plate-like behaviour in the episodic regime. Rigidly moving plates evolve that move with a velocity comparable to the plates on Earth. As subduction is fast, the plates, however, only exist over short timescales. Furthermore we observed further plate-like features such as trench migration and strike-slip motion Stress-dependent viscosity: (b) brittle deformation In a series of experiments we have investigated the effect of a different deformation law. By applying Byerlee s law, we change the rheological model from ductile to brittle behaviour. The results are hardly influenced by this change. In particular, the three regimes (Fig. 10) are also obtained. Like the ductile rheology, the brittle deformation mechanism produces the most plate-like behaviour in the episodic regime. Again periods displaying plate-like motion at the surface are interrupted by long phases in which a stagnant lid grows conductively. We present one example in the following. constant yield stress stagnant lid regime mobile lid regime depth-dep. yield stress σme σ es Fig. 10. Domain regime for the stress-dependent rheology. Transitions from the mobile lid to the episodic regime (σ me ) and from the episodic to the stagnant lid regime (σ es ) are displayed. The parameters used are displayed in the third table of Appendix A. v surf (a) P b (b) time time Fig. 11. Time history of the (a) surface velocity and (b) boundary-size for the episodic regime in case of brittle deformation (run4) Run4 (brittle deformation) The parameters of this run are the same as the ones of run3, except that brittle behaviour was modelled by additionally adopting a depth-dependent yield stress of σ z = Fig. 11 shows the temporal evolution of the surface velocity and the boundary-size P b. The time-averaged surface velocity is v surf = 1.04 cm/yr (Fig. 11a) and the boundary-size P b = 0.06 (Fig. 11b). Compared to the ductile case these values are not as good. This result agrees well with the findings of Tackley (2000b) who also employed a depth-dependent yield stress, however, did not use Byerlee s law. Apparently, a depth-dependent yield stress leads to a loss of plateness, no matter what the specific type of depth dependence is. Generally, temperature- and stress-dependent viscosity convection yields three different regimes. This seems independent of a specific stress dependence as we have shown here for viscoplastic yielding with ductile and brittle deformation and was shown by Tackley (2000b) assuming a different approach.

16 240 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Pressure-dependent viscosity yield stress stagnant lid regime episodic regime mobile lid regime At this stage, our model produces quite a few features of plate tectonics, most of them, however, only appear episodically. In fact, the periods of stagnant lid type are much longer than those characterised by plate motion. We address this effect mainly to the fast downwellings. Anyway a dependence of the viscosity on temperature and stress alone is not realistic for the Earth s mantle, but a further pressure dependence is likely. As it is know from previous work (e.g. Hansen et al., 1993) that an increase of viscosity with pressure (i.e. depth) acts to slow down subduction, this effect can thus potentially influence the episodic behaviour. In a series of experiments we investigated the influence of the pressure dependence on the existence of the regimes. The pressure dependence is varied between RD = 1 and 100. In case of RD = 1 the viscosity does not depend on the pressure, in case of RD = 100 the viscosity increase with depth is a factor of 100. The results are summarised in Fig. 12 (see also the fourth table of Appendix A). In the figure only the results for pressure dependencies up to 30 are shown. Beyond this value the mobile lid regime and the stagnant lid regime are not further separated by an episodic parameter window. With increasing pressure dependence the mobile lid regime prevails. This is plausipressure dependence plate regime ble as pressure dependence reduces the effect of temperature dependence on the viscosity (Schubert et al., 2001). With additional pressure dependence we find that the mobile lid regime can be subdivided into two regions. One region is akin to the mobile lid regime already discussed above. A further part will be called plate regime. This plate regime is obtained for stresses slightly lower than σ me, the yield stress at which the transition from the mobile to the episodic regime occurs. In this regime the episodic behaviour fades away after some time and a steady state is reached. This is characterised by a smoothly moving surface. As in the previous cases, small plates appear for short time intervals in the episodic regime, but with additional pressure dependence we find smoothly evolving plates in the plate regime. We report the detailed findings of such an experiment in the following Run5 (pressure-dependent viscosity) The parameters are identical to the run with a combined temperature- and stress-dependent viscosity ( η = 10 5, Ra = 100, σ 0 = 2, σ z = 0.01). Additionally an exponentially shaped pressure-dependent viscosity variation of a factor of 20 is assumed. This is certainly at the lower end of a realistic variation in the Earth, but allows for an understanding of the principal influence of a depth-dependent rheology on the behaviour of the surface plates. A snapshot of the temperature field is shown in Fig. 13. At this stage the system has already reached a σ me σ es Fig. 12. Domain diagram for the temperature-, stress- and pressure-dependent viscosity. The diagram is spanned by the yield stress and the pressure dependence RD. RD specifies the strength of the exponentially increasing viscosity with pressure. Fig. 13. Snapshot of the temperature field and moving surface (grey area) after transients have died out and a statistical stationary state has been reached.

17 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) statistically steady state. As in the previous cases, upwelling takes place in form of cylindrical plumes while the downwelling forms a sheet-like structure. Here, three upwellings are visible, the fourth one in the back corner cannot be seen. In contrast to stress-dependent viscosity convection, we now find a large-scale component of the convective flow leading to rather extended plates (marked as grey area). Three plates are formed which move relatively to each other. In many respects this simulation gives the best results as far as plateness is concerned. This can be seen in Table 1 from the time-averaged diagnostic values. Compared to the previous run in the mobile lid regime (run2), we find that plate boundaries are stronger localised ( P b = 0.11), whereas the amount of plates has increased ( P p = 0.89). Furthermore, the plates move as almost perfect rigid entities, the internal deformation is only ɛ plate = This is much lower than for plates in the mobile lid regime without pressure dependence. The ratio of toroidal to poloidal kinetic energy is similar to that of the mobile lid regime shown before, but it is reduced ( R TP = 0.08) compared to the value that appeared in the episodic regime at times when plates existed (R TP = 0.17). An explanation for this phenomenon is provided by the findings of Dumoulin et al. (1998). They report that toroidal motion is closer associated with strike-slip motion than with rotation of plates as a whole. Since we hardly find strike-slip motion here this is consistent with our result. A downside is the drastic reduction in the plate velocity ( v plate = 0.05 cm/yr), which, however, leads to a more continuous plate motion. The continuous behaviour is achieved by slow subduction. As pressure dependence leads to an increase of viscosity with depth, the sinking slab runs against resistance. Subduction is slow (v d = 35.70, Table 1). The velocity of the upwelling is v u = In comparison to cases without a pressure-dependent viscosity we find a reduction of the up- and downflow velocity. The reduction of the subduction speed is, however, more pronounced. In Fig. 14 the time-averaged surface velocity, the ratio of boundary to plates and the toroidal poloidal energy ratio as a function of the pressure-dependent viscosity are presented. Obviously, additional pressure dependence strongly reduces the plate velocity. Although there is a strong variation in the value of the diagnostic value pressure dependence V surf + Rp R TP Fig. 14. Diagnostic values in dependence on the pressure-dependent viscosity. The input parameters were: η = 10 5, Ra = 200 and σ 0 = 4. toroidal poloidal ratio, we find that the ratio is generally slightly reduced. Moreover pressure dependence leads to a large-scale flow with extended plates and a reduced boundary plate ratio. Another interesting feature appearing self-consistently once pressure dependence of the viscosity is employed, is the formation of a low-viscosity zone. In Fig. 15 the horizontally averaged viscosity over depth z stagnant lid regime mobile lid regime plate regime 1e horizontally-averaged viscosity Fig. 15. Viscosity-depth profile for an example of the stagnant lid regime, the mobile lid regime and the plate regime. All examples were conducted with the same amount of pressure dependence. The input parameters were: η = 10 5, Ra = 100, RD = 20 and σ 0 = 0.1 (mobile lid regime), σ 0 = 3 (plate regime), σ 0 = 100 (stagnant lid regime).

18 242 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) is shown for three cases: the stagnant lid regime, the mobile lid regime and the plate regime. For all cases the same amount of pressure dependence was assumed, so that only the value of the yield stress is different. In the mobile lid regime, the stress dependence overcomes the temperature dependence and thus the surface moves fluid-like. The viscosity at the top and bottom boundary are almost identical. The viscosity profile of the stagnant lid regime is virtually identical to the one found in thermoviscous convection. A significant viscosity variation occurs in the cold boundary layer. Below the stagnant lid viscosity is almost constant. In the plate regime the viscosity on top is comparable to that in the stagnant lid regime. At shallow depth the viscosity strongly decreases locally. The appearance of plate-like behaviour thus goes hand in hand with the formation of a low-viscosity zone below the upper surface. To our knowledge, the self-consistent formation of an asthenosphere is reported here for the first time. We have shown that the pressure dependence is not the only important factor necessary for the formation of a low-viscosity zone. All cases presented considered the same amount of pressure dependence, but only in the plate regime an asthenosphere was observed. Interestingly, the cases of the mobile lid regime and the plate regime display a viscosity maximum in the mantle which is qualitatively akin to the finding of Forte and Mitrovica (2001). From earlier studies on convection with plates (Lowman et al., 2001) it is known that the convective pattern is strongly influenced by the aspect ratio of the computational domain. Especially convection with a pressure-dependent viscosity has been demonstrated to induce long wavelength flow patterns (e.g. Balachandar et al., 1992; Hansen et al., 1993; Bunge et al., 1997; Tackley, 2000c) which would be suppressed in small aspect ratio domains. As calculations in a fully self-consistent model approach are computationally very demanding, in this work we restricted ourselves to an aspect ratio of 2. This allowed for a wide range of parameter runs in an acceptable time. But in order to assess the influence of the aspect ratio, we conducted one run in an aspect ratio 4 domain. The results of this run will be compared to the results of a run conducted under identical conditions in a box of aspect ratio 2. The parameters were a temperature dependence of η = 10 5, a stress dependence of σ 0 = 2, a pressure dependence of RD = 5 and a Rayleigh number of Ra = 200. Under these conditions in both geometries a steady state developed, displaying continuous plate-like motion. Fig. 16 shows snapshots of the temperature fields for both cases. In both calculations a steady flow evolves for the chosen parameter combination. The flow pattern is characterised by cylindrical upwellings, while the downflow takes place in a sheet-like manner. The diagnostic values (Table 2) reveal virtually very similar results, although small differences in the plate deformation are observed. While in the aspect ratio 2 case only one plate results, three plates have formed in the aspect ratio 4 case, which are sketched in the small plot in Fig. 16a. This also results in a higher ratio of toroidal to poloidal kinetic energy, as strike-slip motion is enhanced. Fig. 16. Snapshots of the temperature field for a model calculation in a box of (a) aspect ratio of 4 and (b) in a box of aspect ratio of 2.

19 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Table 2 Comparison of the calculated diagnostic values for two runs with different aspect ratios Diagnostic values v plate ɛ plate P b P p R TP v u v d Aspect ratio Aspect ratio The over bar values are the time-averaged values, the others the ones at one point of time. One further difference is displayed in Fig. 17 where the distribution of the vertical velocities at mid-depth are shown for the same time instants as portrayed in Fig. 16. The highest positive values correlate with the observed upwellings and subduction correlates with the minima of the isosurface. Subduction is obviously slowed down when taking place at the edge of the domain. The downflow in the middle of the aspect ratio 4 domain reaches almost double the vertical velocity as found for the downflow in the aspect ratio 2 domain which is located at the edge. The general finding that subduction is slowed down by a pressure-dependent viscosity, thus leading to a smooth (even steady) temporal evolution of plates, clearly also holds for the aspect ratio 4 domain. As pointed out in Lowman et al. (2001) we underline the finding that the dynamics of the coupled convection-plate system is influenced by the aspect ratio, that however fundamental aspects of the self-consistent origin and evolution of plates can be investigated in a reasonable approximation in the small aspect ratio domain. Applying larger aspect ratios also helps to diminish the effect of the side walls. In all our experiments reflective boundaries have been considered and they do certainly influence the flow pattern. One effect ob- served is the bending backwards of the subducted material when coinciding with the boundary. The same phenomenon, however, in the middle of the box if the subduction zone moves against a piece of the stagnant lid and thus the retrograde bending is not exclusively a feature caused by the reflective boundary condition. Interestingly the downflow develops a geometry as observed on Earth (Van der Voo et al., 1999). A further point of discussion is the model geometry. Our Cartesian domain certainly faces severe limitations. A fully spherical geometry would definitely be more realistic but at this stage spherical models capable of handling strong lateral viscosity variations are just under development. Although, we cannot speculate on the possible results of spherical models, we feel that in the meantime important aspects of plates can be explored by Cartesian models. First results in a sphere (Zhong et al., 2000; Monnereau and Quéré, 2001) indicate in fact close similarities between the Cartesian and spherical model approaches. The models do differ with respect to the mean temperatures Cartesian models yield a higher mean temperature. In order to obtain self-consistent plate-like behaviour in spherical models, one would expect that only the rheological parameters need to be properly adjusted. Fig. 17. Distribution of the vertical velocity for two different aspect ratio calculations. (a) Aspect ratio 4 result and (b) result for the example with the aspect ratio 2.

20 244 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Summarising, pressure dependence yields a subdivision of the mobile lid regime into the mobile lid with a fluid-like surface and into the plate regime with smoothly evolving plates. The best plate-like behaviour so far is obtained in the plate regime. This is bordered by the mobile lid regime and the episodic regime. In the plate regime extended plates move rigidly and continuous in time. Furthermore, the formation of a low-viscosity zone appears self-consistently in this regime Internal heating The planform of convection is known to be dependent on the amount of internal heating (Turcotte and Schubert, 1982). Most studies which have taken into account a combination of internal and bottom heating have specified the bottom heat flux rather than the temperature, and mainly assumed adiabatic conditions at the lower boundary (Houseman, 1988; Tackley, 1998, 2000b,c). Thermal conditions at the core mantle boundary are certainly better represented by a fixed temperature, rather than by a vanishing heat flux. In this study, we have therefore employed a constant temperature at the lower boundary together with an internal heating rate. This was also done by Weinstein et al. (1989), Weinstein and Olson (1990), Hansen et al. (1993), Sotin and Labrosse (1999) and Lowman et al. (2001). Internal heating enhances the vigour of the downflow (McKenzie et al., 1974) and can as such potentially lead to an increase of the plate velocity. An overall view of the influence of internal heating on the convective regimes is given in Fig. 18. Itbecomes obvious that the transition from one regime to the other occurs at lower values of the yield stress if the amount of internal heating is increased. At sufficiently high internal heating rates, only the stagnant lid regime would be obtained. This is a consequence of the higher temperature due to internal heating. Under these circumstances the temperature dependence dominates the behaviour rather than the stress dependence and this leads to the stagnant lid regime. Like in the bottom heated cases, the pressure dependence of the viscosity leads to a subdivision of the mobile lid regime. As for bottom heated cases with a viscosity depending on pressure, we also find in case of internal heating that the best plate-like behaviour occurs in a part yield stress episodic regime plate regime mobile lid regime stagnant lid regime internal heating σ me σ es Fig. 18. Appearance of the convective regimes for different rates of internal heating. For information on the data set see Appendix A. of the mobile lid regime. In this plate regime plate motion appears continuously and extended plates result. An example from this regime, which will be described subsequently, has been carried out under the same conditions as the previous run with additional internal heating Run6 (internal heating) The rate of internal heating was chosen as Q = 1, which leads to an amount of internal heating that is 10% of the overall heat production. Fig. 19 exhibits the corresponding temperature field. That part of the surface which moves as a plate (according to our plate-criterion) is pictured as a grey area covering almost the entire surface. Interestingly, the form of the upwelling has changed from a cylindrical to a sheet-like structure. Recent seismological studies (Ni et al., 2002) indicate the existence of a largely two-dimensional upwelling beneath southern Africa. It is tempting to interpret the sheet-like structure as a result of internal heating, but its appearance is a consequence of the combined influence of internal heating, pressure dependence of the viscosity and the presence of plates. At this stage we cannot delineate the precise relationship. The time history of the surface velocity and the boundary-size is displayed in Fig. 20. This figure displays a characteristical feature of the plate regime. At first several overturns are visible but finally a steady

21 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) Fig. 21. Distribution of the vertical velocity in the plane z = 0.5 for the case shown in Fig. 19. Fig. 19. Snapshot of the temperature distribution for the internally heated case (run6). Here a stationary flow is reached. The moving surface is marked as grey area. state is reached. Once the steady state is reached, the surface velocity (v surf = 0.07 cm/yr) has slightly increased, as compared to the purely bottom heated case. While internal heating changes the surface velocity, it does not influence the internal deformation (ɛ plate = ) much. Applying our plate-criterion yields a time-averaged boundary-size of P b = 0.16, so that 16% of the surface is given by boundaries and 84% are covered by plates. The ratio of toroidal to poloidal energy (dashed line in Fig. 20) is, in this case, close to zero, at least when the resulting stationary state is reached. This is due to diagnostic value v surf P b R TP time Fig. 20. Surface velocity (solid line), boundary-size (dotted line) and toroidal poloidal ratio (dashed line) as function of time for the case with additional internal heating. the highly symmetric pattern of the flow. A similar result was reported by Weinstein (1998) who also found a decrease in the toroidal component, if the pattern shows a high degree of symmetry. In his study, however, internal heating generally leads to asymmetry, while here we present at least one case where symmetry prevails despite internal heating. An analysis of the strength of the downflow versus upflow confirms the expected result, i.e. internal heating strengthens the downflow and weakens the upflow. Fig. 21 portrays the distribution of the vertical velocity at mid-depth, i.e. at z = 0.5. We find the maximum downward velocity of v d = to be higher than that for the solely bottom heated case (v d = 35.70), while the maximum upward velocity is with v u = significantly smaller than that in the bottom heated case (v u = ). Although the upwelling is slower in this case, the surface velocity has increased. This clearly shows the strong influence of the downwelling on the plate velocity. It is obviously the downflow which determines the surface velocity. The general behaviour of the surface velocity, the ratio of boundary to plate and the ratio of toroidal to poloidal kinetic energy in dependence on the amount of internal heating is summarised in Fig. 22. As presented for one specific run, we generally find an increase in the surface velocity with increasing internal heating. The ratio of boundary to plate is almost constant, signifying that the size of the plates is not much influenced by internal heating. Due to a highly symmetric planform in some cases, the ratio of toroidal to poloidal energy is significantly reduced. We put this down mainly to geometrical effects, i.e. the small aspect ratio, rather than being a typical effect of internal heating.

22 246 C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) diagnostic value o e-06 1e-10 V surf Rp R TP + 1e internal heating Fig. 22. Surface velocity, boundary plate ratio and toroidal poloidal kinetic energy as function of internal heating. Input parameters: η = 10 5, Ra = 200, RD = 10 and σ 0 = Depth-dependent thermal expansivity Laboratory experiments have clearly revealed a decrease of the thermal expansion coefficient throughout the mantle (Chopelas and Boehler, 1989, 1992). As demonstrated by Hansen et al. (1991) the decrease of the expansion coefficient influences the thermal structure and the pattern of convection in a similar way as pressure-dependent viscosity does. For an increasing depth dependence of the thermal expansivity (1 < α<1/7), the transitions to the different regimes occur at lower yield stresses as shown in Fig. 23. Here the thermal expansivity decreases across the mantle by a factor of 1 7. If the thermal expansivity is strongly depth-dependent, the stagnant lid regime prevails, because the temperature increases strongly with depth and therefore gains influence. This is comparable to an increased amount of internal heating (Fig. 18) Run7 (expansivity + pressure) In order to clarify the effect of thermal expansivity with respect to plate-like behaviour we have conducted a further experiment in which the thermal expansion coefficient was assumed to decrease by a factor of 3 throughout the mantle. Otherwise the model parameters were the same as in the previous case with one further exception. We had to lower the value of the yield stress slightly in order to get into the plate regime. Altogether the parameters of this run are: η = 10 5, Ra = 100, σ 0 = 1.5, σ z = 0.01, RD = 20, Q = 1 and α = 1/3. This run shows exemplarily, that again a stationary state is reached. Its temperature distribution is visualised in Fig. 24. A strong upwelling has formed in the centre with downwellings along the edges. Remarkably, this run also shows that finally a sheet-like upwelling develops. Differently from the previous case the upwelling develops in the middle of the box. The formation of a sheet-like upwelling seems, therefore, not to be a pure edge-effect. The pressure dependence of the viscosity and the thermal expansivity lead to a large-scale flow resulting in rather extended plates at the surface (marked by grey areas). The extension of Fig. 23. Regime diagram spanned by the yield stress and depth-dependent expansivity. The parameters used for conducting the different runs are listed in Appendix A. Fig. 24. Snapshot of the temperature field for the run with variable thermal expansivity (run7). The grey area marks the moving surface.