Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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.
© 2004 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.: +49-251-8333597;
fax: +49-251-8336100.
E-mail address: [email protected] (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).
0031-9201/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.pepi.2004.01.006
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C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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 64 × 64 × 32
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
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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.
227
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 6.
2. 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,
∂t
(1)
∇ · u = 0,
(2)
−∇p + ∇ · σ + RaTez = 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:
∂uj
∂ui
σij = 2ηėij = η
(4)
+
∂xj
∂xi
with η the dynamic viscosity and ėij the strain-rate
tensor. ez is the unit vector in z-direction and Q is the
rate of internal heating:
Q=
RaH
Hd2
=
Ra
kT
(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. RaH 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.
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C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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 =
α0 gρ0 Td3
,
η0 κ
(6)
which is defined by using the thermal expansivity
α0 , the viscosity η0 and the density ρ0 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 64 × 64 × 32. Comparisons with
single experiments performed on a 128×128×64 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 0.92. 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:
ηT = exp(−rT),
(8)
where r = ln(∆η) and η = η0 /ηT being the viscosity contrast over the domain, with η0 = 1 the viscosity at the surface (for T = 0) and ηT 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
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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:
η=
2
1/ηT + 1/ησ
(9)
with the temperature-dependent viscosity ηT specified
in Eq. (8) and a stress-dependent part ησ :
σY (z)
.
ησ = η∗ +
( : )1/2
(10)
σ Y (z) is the yield stress, ( : )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):
σY (z) = σ0 + σz (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
229
yield stress. Thus in case of σz = 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:
ηp,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:
3
α0
α(z) =
(14)
m(1 − z) + 1
with α0 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-
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C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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.
4.1. Surface velocity vsurf
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:
vsurf := u2 + v2 .
(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 m2 /s is the thermal diffusivity
and d the height of the domain which is, for the whole
mantle convection, d = 2900 km.
we do not only use the surface deformation but also
the horizontal divergence:
∂u ∂v
+ ,
(17)
∇h · u =
∂x ∂y
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 (Pb ) when more than 20% of
the maximum deformation ( > plate ) and more than
20% of the maximum absolute divergence is reached.
The rest is made up by either the stagnant lid (Ps ), if
surface velocities are low, or by plates (Pp ) when 80%
of the maximum surface velocity (vplate = 0.8vmax
surf ) is
reached.
4.4. Boundary–plate ratio Rp
4.2. Surface deformation surf
The surface deformation is determined by the square
root of the second invariant of the strain-rate tensor at
the surface given by:
2
1/2
∂u 2
∂v
1 ∂u ∂v 2
surf :=
+
.
+
+
∂x
∂y
2 ∂y
∂x
(16)
According to Tackley (2000b), regions displaying less
or equal plate := 0.2max
surf are regarded as plates while
regions with more than 20% of the maximum surface deformation are considered as plate boundaries.
If plate is very low, we thus have a rigid surface (i.e.
plate) but for a high plate the surface is strongly deformed, i.e. fluid-like.
4.3. Plate-criterion
Besides exhibiting little deformation, plates should
also move with an almost uniform velocity. Therefore,
According to the plate-criterion we can determine the ratio of boundaries to plates, which is the
boundary-size Pb divided by the plate-size Pp . 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
Rp ≈ 0.18.
4.5. Toroidal–poloidal ratio RTP
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;
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
Olson and Bercovici, 1991; Ribe, 1992; LithgowBertelloni 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:
uh = uh,pol + uh,tor = ∇ h Φ + ∇ h × (Ψ ez ).
(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:
∇h2 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 × Ψ ez ) = −∇h2 Ψ ez = (∇ × u)h .
(20)
The ratio of toroidal–poloidal energy in the surface is
finally given as:
(∇h Ψ)2 RTP =
(21)
(∇h Φ)2 with Ψ the toroidal and Φ the poloidal component.
4.6. Plume velocity vu or vd
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 vu is determined as the maximum velocity
in this plane and the velocity of the downwelling vd
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
231
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 vplate is about a few
cm per year and the ratio of toroidal to poloidal energy RTP is between 0.25 and 0.5 (Lithgow-Bertelloni
et al., 1993). The values to be expected for the deformation of the plate plate , the boundary-size Pb
and the plate-size Pp are illustrated by some case
studies. For isoviscous convection (i.e. a fluid-like
surface) with a Rayleigh number of Ra = 107 we
find a plate deformation of plate = 7957.46. For
thermoviscous convection with a viscosity contrast
of η = 105 and a top Rayleigh number of 100
(i.e. a bottom Rayleigh number of 107 ) the surface
is highly viscous and a rigid lid covers the complete surface. In this case the deformation of the
plate is 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 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
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C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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
σ0
Run1
Run2
Run3
Run4
Run5
Run6
Run7
σz
10
0.1
2
2
2
2
1.5
0
0
0
0.01
0.01
0.01
0.01
Diagnostic values
RD
1
1
1
1
20
20
20
α
1
1
1
1
1
1
1/3
Q
0
0
0
0
0
1
1
v̄plate
3.2 ×
1.42
1.30
1.04
0.05
0.07
0.04
plate
10−4
m
4 sin([2n − 1]πx)
,
π
2n − 1
0.8
6624.58
2153.98
5936.52
270.6
131.88
154.9
0
0.18
0.05
0.06
0.11
0.16
0.13
P̄p
0
0.82
0.28
0.42
0.89
0.84
0.87
R̄TP
4.3
1.5
6.9
5.1
8.3
0.3
1.1
×
×
×
×
×
×
×
10−4
10−2
10−2
10−2
10−2
10−14
10−2
vu
vd
–
–
1867.90
3079.88
493.36
196.05
277.72
–
–
−1349.05
−3819.72
−35.70
−54.03
−38.10
plate-size to Pp = 0.34. For m = 10 we find Pb =
0.08 and Pp = 0.92. For m = 500 two ideal plates
(Pp = 0.97) with a strongly localised boundary (Pb =
0.03) are observed.
two plates:
f(x, y) =
P̄b
0 ≤ x, y ≤ 2.
n=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 Pb = 0.84 and a plate-size of Pp = 0.16. 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 Pb = 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 = 1 a
sine function results, for m = 500 the boxcar function represents
two rigidly moving plates with an infinitely narrow boundary.
5.1. 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-
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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 < 102 ) thermoviscous convection resembles isoviscous convection, so
that no plate-like behaviour is found. In the transitional
regime (102 < η < 104 ) 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 > 104 ) 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.
5.2. Stress-dependent viscosity: (a) ductile
deformation
The rigid surface formed as a result of a viscosity contrast of η = 105 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
233
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
234
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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 = 105
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.
5.2.1. Run1 (stagnant lid regime)
Fig. 3 shows a snapshot of the temperature field as
obtained with a high yield stress (parameters: η =
105 , Ra = 100, σ0 = 10 and σz = 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.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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 (RTP = 4.3×10−4 ).
Expectedly, the plate-criterion gives a value of Ps = 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.
5.2.2. Run2 (mobile lid regime)
Reducing the value of the yield stress from σ0 =
10 to σ0 = 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 = 105 , Ra = 100 and σ0 =
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 (Tint = 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 (Pb = 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
(plate = 6624.58). This value is similar to the deformation found for isoviscous convection with the same
bottom Rayleigh number (iso
plate = 7957.46). Thus the
surface behaves rather like a fluid than as plates. The
surface moves relatively fast (vmax = 1.78 cm/yr).
The ratio of toroidal to poloidal energy (RTP =
1.5 × 10−2 ) is slightly increased because the surface
behaves fluid-like and thus shear motion can appear.
235
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.
5.2.3. 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 = 105 , Ra = 100,
σ0 = 2 and σz = 0. The yield stress σ0 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
236
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
Fig. 5. Temperature distribution for one complete overturn for a run with ductile deformation (run3). The grey area illustrates the uniformly
moving surface.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
237
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 P̄p and the boundary-size P̄b , 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
v̄plate = 1.3 cm/yr which lies within realistic limits.
Furthermore, the plates are less internally deformed
(¯plate = 2153.98) than they were in the mobile lid
regime (¯plate = 6624.58). 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
0.12
(d)(e)
(f)
(c)
0.08
(b)(g)
0.04
(h)
(i)
0.16
RTP
0
(a)
0
0.2
0.4
0.6
0.8
1 (j) 1.2
1.4
time
Fig. 7. Ratio of toroidal–poloidal energy RTP as a function of
time. The dots mark time instants displayed in Fig. 5.
238
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
300
- 300
-1e+03
1e+03
Vz
2000
1000
0
−1000
2
1.6
1.2
x
0.8
0.4
02
1.6
1.2
0.8
0.4
0
y
Fig. 8. Distribution of the vertical velocity at depth z = 0.5 for
the case shown in Fig. 5c.
diagnostic value
the remaining stagnant lid—a feature which resembles strike-slip motion. A maximum value of RTP =
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 (vu = 1867.90)
+
o
1
∆
0.01
∆
Vsurf +
Rp
+
o
+ +
where the upwelling breaks through the surface and
the highest negative value (vd = −1349.05) 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 (vu = 1969.38) but reduced
downflow velocity (vd = −49.52) resulted in a likewise reduced surface velocity (v̄plate = 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 (vsurf small)
covers the whole surface (Ps = 1, i.e. Rp = 0).
As neither strike-slip motion nor rotation appears
∆
R TP
∆
+
+
+
∆
∆
0.0001
1
∆
10
100
1000
yield stress
Fig. 9. Diagnostic values (surface velocity vsurf , boundary–plate ratio Rp and ratio of toroidal–poloidal kinetic energy RTP ) in dependence
on the yield stress. The input parameters of the runs were: η = 105 and Ra = 200.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
in the surface, the toroidal–poloidal ratio RTP 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.
239
4
3
vsurf
2
1
0
0
0.4
0.8
1.2
1.6
2
1.2
1.6
2
time
(a)
1
5.3. Stress-dependent viscosity: (b) brittle
deformation
0.8
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
10
stagnant lid regime
1
0.1
mobile lid regime
0.01
0.001
0.001
1
100
depth-dep. yield stress
σ me
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.
Pb
0.6
0.4
0.2
0
(b)
0
0.4
0.8
time
Fig. 11. Time history of the (a) surface velocity and (b) boundary-size for the episodic regime in case of brittle deformation
(run4).
5.3.1. 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 σz = 0.01.
Fig. 11 shows the temporal evolution of the surface
velocity and the boundary-size Pb . The time-averaged
surface velocity is v̄surf = 1.04 cm/yr (Fig. 11a) and
the boundary-size P̄b = 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.
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C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
5.4. Pressure-dependent viscosity
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 plausi12
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.
5.4.1. Run5 (pressure-dependent viscosity)
The parameters are identical to the run with a combined temperature- and stress-dependent viscosity
(η = 105 , Ra = 100, σ0 = 2, σz = 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
yield stress
10
stagnant lid regime
8
6
c
episodi
4
regime
2
0
plate regime
mobile lid regime
0
5
10
15
20
25
30
pressure dependence
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.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
10
diagnostic value
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 (P̄b = 0.11), whereas the amount
of plates has increased (P̄p = 0.89). Furthermore, the
plates move as almost perfect rigid entities, the internal deformation is only plate = 270.6. 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 (R̄TP = 0.08) compared to the value
that appeared in the episodic regime at times when
plates existed (RTP = 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 (v̄plate = 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 (vd = −35.70, Table 1). The velocity
of the upwelling is vu = 493.36. 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
241
1
0.1 +
∆
+
+
+
0.01
+
∆
∆
∆
0.001 ∆
0.0001
+
∆
∆
10
100
pressure dependence
Vsurf
Rp
+
R TP
∆
Fig. 14. Diagnostic values in dependence on the pressure-dependent
viscosity. The input parameters were: η = 105 , Ra = 200 and
σ0 = 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
1
0.8
0.6
z
0.4
stagnant lid regime
mobile lid regime
plate regime
0.2
0
1e-05
0.001
0.1
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 = 105 , Ra = 100, RD = 20 and
σ0 = 0.1 (mobile lid regime), σ0 = 3 (plate regime), σ0 = 100
(stagnant lid regime).
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C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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 = 105 , a
stress dependence of σ0 = 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.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
243
Table 2
Comparison of the calculated diagnostic values for two runs with different aspect ratios
Diagnostic values
Aspect ratio 4
Aspect ratio 2
v̄plate
plate
P̄b
P̄p
R̄TP
vu
vd
0.190
0.142
682.1
380.3
0.12
0.19
0.88
0.81
0.0129
0.0032
1124.24
1133.89
−300.18
−173.04
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.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
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.
5.5. 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. It becomes 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
7
yield stress
244
iso
dic
plat
e re
3
1
stagnant lid regime
ep
5
gim
reg
im
e
e
mobile lid regime
0
2
4
6
8
10
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.
5.5.1. 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
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
245
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 (vsurf = 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 (plate =
131.88) much. Applying our plate-criterion yields a
time-averaged boundary-size of P̄b = 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
1
vsurf
Pb
RTP
0.8
0.6
0.4
0.2
0
0
0.4
0.8
1.2
1.6
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 vd = −54.03 to be higher than
that for the solely bottom heated case (vd = −35.70),
while the maximum upward velocity is with vu =
196.05 significantly smaller than that in the bottom
heated case (vu = 493.36). 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.
246
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
1
diagnostic value
0.01
+
+
∆
∆
o+
+
+
∆
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).
1e-06
Vsurf
Rp
1e-10
1e-14
+
R TP
0
∆
1
∆
2
∆
3
internal heating
Fig. 22. Surface velocity, boundary–plate ratio and toroidal–
poloidal kinetic energy as function of internal heating. Input parameters: η = 105 , Ra = 200, RD = 10 and σ0 = 2.
5.6. 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
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.
5.6.1. 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 = 105 ,
Ra = 100, σ0 = 1.5, σz = 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. 24. Snapshot of the temperature field for the run with variable thermal expansivity (run7). The grey area marks the moving
surface.
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
0.6
Vz
0.4
0.3
110
50
−21
250
150
50
−50
vsurf
Pb
RTP
0.5
diagnostic value
247
0
0.2
0.4
0.8
x
0.1
0
0
0.5
1
1.5
2
0.4
2 0
1.2
1.6
2
y
1
diagnostic value
the flow is likely to be limited by the size of the box,
so that a box of larger aspect ratio would probably lead
to a flow and plates of larger horizontal extension.
The stationary character of the flow is evident from
the time history plot of the surface velocity and the
amount of surface deformation (Fig. 25). As compared
to the previous run the surface velocity is slightly
reduced (vsurf = 0.04 cm/yr). The amount of plate
boundary is also slightly reduced (Pb = 0.13), which
shows that the plate boundaries are strongly localised.
The toroidal–poloidal energy ratio (dashed line in
Fig. 25) is increased compared to the previous run.
This is due to the less symmetric nature of the flow.
A more careful inspection of Fig. 24 shows that the
upwelling does not extend through the whole box, thus
an asymmetric structure evolves which leads to the
increase in the toroidal energy.
As the surface velocity of this run is reduced compared to the run before, we would also expect slow
subduction and this is in fact what we observe (vd =
−38.10) (Fig. 26). The influence of the downwelling
seems to be the most important factor as we have a
decreased surface velocity even though the velocity of
the upwelling plume (vu = 277.72) is higher than that
of the previous run.
The general influence of a depth-dependent expansivity on the surface velocity, the ratio of boundary
to plate and the toroidal–poloidal energy ratio is presented in Fig. 27. The behaviour is comparable to the
behaviour that results from a decrease of the viscosity
as a function of pressure. The plates become slightly
1.6
0.8
Fig. 26. Distribution of the vertical velocity (at z = 0.5).
time
Fig. 25. Temporal evolution of the surface velocity (solid line),
boundary-size (dotted line) and toroidal–poloidal kinetic energy
ratio (dashed line) for depth-dependent thermal expansivity.
1.2
0.1+
+
+
+ +
+
0.01
0.001∆
∆
∆
∆
∆
∆
∆
0.0001
1
depth−dep. expansivity (1/ ∆α )
Vsurf
Rp
+
R TP
∆
10
Fig. 27. Diagnostic values as function of the depth dependence
of the thermal expansivity. Parameters: η = 105 , Ra = 200,
RD = 10 and σ0 = 2.
more extended, so that the boundary–plate ratio Rp =
Pb /Pp decreases. This, however, goes hand in hand
with a lower ratio of toroidal to poloidal kinetic energy because the possibility of strike-slip motion is reduced. Furthermore, we presented that in the case of
a depth-dependent expansivity subduction is slowed
down, which leads to an overall decrease in the plate
velocity.
6. Conclusion
The main aim of the paper was to investigate configurations which allow for the self-consistent formation
of plates in a model of mantle convection. We have
demonstrated that a combination of temperature- and
248
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
stress-dependent viscosity in fact leads to plate-like
behaviour, once the value of the yield stress is chosen appropriately. The application of strict criteria for
the plate-like behaviour, i.e. the plate moves virtually
undeformed and all deformation occurs at the plate
boundary, revealed a ‘plateness’ of excellent quality.
Furthermore, the presence of the plates resulted in the
excitation of toroidal motion which made up to about
20% of the total energy of motion. But plate motion
appeared only within short periods. During these periods we find that most of the surface still consists of
the stagnant lid. The source of the episodic behaviour
is the unequal strength of the up- and downwelling
currents. Sheet-like downwellings subduct the plate so
fast, that material provided by cylindrical upwellings
cannot form new plates at the same rate. Thus a cyclic
behaviour characterised by (a) the existence of a stagnant lid, (b) the initiation of plate tectonics, (c) the
subduction of the plate and (d) the conductive growth
of a new lid which leads back to the initial situation
results. This type of behaviour was observed for both
ductile and brittle rheology.
The parameter regime in which the episodic plate
motion takes place is bordered by two other regimes.
For higher values of the yield stress we observed a
stagnant lid type of flow as is characteristic for only
temperature-dependent viscosity. At lower values of
the yield stress no plate motion was found either because the surface was in fact mobile but also strongly
deformed through the motion. In this regime, the
mobile lid regime, the surface behaves like a fluid
rather than like a plate and the type of flow resembles
that of isoviscous convection. Thus, for temperatureand stress-dependent rheology plate motion appeared
exclusively in the episodic regime, no matter if ductile or brittle behaviour was considered. Interestingly,
we sometimes observed a smooth change from the
episodic to the stagnant lid regime, i.e. episodic plate
motion showed up transiently but finally the system
ended in the stagnant lid type of flow. A change from
one regime to another can potentially mark a turning
point during the evolution of a planet. It has been
speculated that Mars exhibited plate tectonics in its
early days while it is presently covered by a stagnant
lid (Sleep, 1994). Such a behaviour can be explained
by the observed change.
Taking further a pressure dependence of the viscosity into account leads to an important change in
the style of the flow towards a continuous motion
of extended plates. We again observed the existence
of the three regimes: mobile lid, episodic and stagnant lid regime. This time, however, plate-like motion did not only show up in the episodic regime,
but in a small window of yield stresses the mobile
lid regime changed its character. Thus the mobile lid
regime can be subdivided into a part which yields a
flow as is characteristic for isoviscous convection and
a part that shows continuous plate-like behaviour. In
this regime a steady state is reached, while in the mobile lid regime the flow is time-dependent. This plate
regime is bordered by the mobile lid regime for low
yield stresses and the episodic regime for higher yield
stresses. Within the plate regime we found continuous
plate motion, because the increase of viscosity with
pressure (i.e. with depth) acts to weaken the downflows. Therefore the dominance of subduction over
the upwellings is diminished. New surface can thus be
created while the already existing plate is subducting.
This parameter combination, i.e. a temperature- and
stress-dependent viscosity which moreover increases
with depth, altogether exhibited very satisfactory results as far as the plate-diagnostics are concerned. The
plates were extended and moved rigidly, but the ratio of toroidal to poloidal energy decreased with this
rheology which we consider to be a downside. We believe that the main reasons for the low ratio are the
extended plates and the relatively small aspect ratio of
the domain, because they prevent the development of
strike-slip motion. This view is supported by an experiment in a domain of aspect ratio 4. Here, strike-slip
motion was more prominent, thus leading to an increase of the ratio of toroidal to poloidal energy. Furthermore an additional pressure dependence leads to
the self-consistent appearance of a low-viscosity zone.
This phenomenon, however, was only observed in the
parameter region where the system exhibits plate-like
motion and is not simply due to pressure-dependent
viscosity.
In a further step we examined the influence of internal heating on plate behaviour. Rather than combing it with an adiabatic condition we assumed a fixed
temperature at the lower boundary. We feel that this
scenario matches more realistic conditions since the
core–mantle boundary seems to be better represented
by specifying a constant temperature, rather than assuming a vanishing heat flux. In our experiments inter-
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
nal heating leads to a highly symmetric situation and
therefore to an even lower ratio of toroidal to poloidal
energy. This is certainly not a general feature of internally heated flow but in our case probably the result
of the small computational domain. As again a pressure dependence was considered, plates did not only
develop in the episodic regime, but in a small window
of yield stresses even a stationary state of the flow was
received. It is clear that plate tectonics on Earth is not
stationary but we feel it is important to show which
wide range of evolutionary branches can dynamically
be realised. A further noteworthy feature is the appearance of a virtually sheet-like upwelling. While in general for temperature-, stress- and pressure-dependent
viscosity rising plumes form cylindrical structures, we
noticed the change towards a sheet-like structure, once
internal heating was introduced. This is particularly
interesting since seismic studies have revealed the existence of such upwellings beneath Africa (Ni et al.,
2002). In a few cases we also found sheet-like upwellings without internal heating. At the moment we
thus cannot clearly decide on the mechanism being responsible for the change from cylindrical to sheet-like
plumes. It seems that the pressure dependence and internal heating and especially the presence of plates are
of key importance.
Besides the pressure dependence of the viscosity,
laboratory experiments suggest a decrease of the thermal expansivity with pressure (Chopelas and Boehler,
1989, 1992). In order to mimic this effect we assumed the thermal expansivity to decrease by a factor of 3 throughout our model domain. We observed
two main effects: (1) the downflow is weakened and
with this the surface velocity is slightly reduced and
(2) an increase in the plate-size. These effects are
similar to the effects arising from a depth-dependent
viscosity.
Interestingly, in all cases displaying plate-like
surface motion, the velocity of the surface seemed
widely controlled by the speed of subduction. The
upwellings obviously play only a minor role. This
is a conclusion from our observation that depth dependence of properties tends to strengthen the upflow and to weaken the downflow, which clearly
results in the reduction of the plate velocity. On
the other hand, internal heating weakens the uprising plumes to the advantage of the sinking currents, which results in an increase of plate veloc-
249
ities. This, altogether, supports the view of Bird
(1998) and Schubert et al. (2001) that the plate velocity is much more controlled by the descending mantle
currents than by upwellings.
How meaningful are our investigations for planetary
dynamics, in particular for the Earth? Admittedly, our
model is just a coarse approximation of reality, especially with respect to the geometry. We have demonstrated that the principal behaviour of the model is not
altered when larger aspect ratios are employed and
this finding indicates that similar phenomena will take
place in a spherical geometry. Certainly, many detailed
features of mantle convection are left out. But given
the fact that the integrated system of plates and mantle convection is still poorly understood, we consider
it sensible to first identify key ingredients allowing
for the self-consistent evolution of plate tectonics in a
model of mantle convection.
Acknowledgements
This work is supported by the Deutsche Forschungsgemeinschaft (grant number Ha1765/8-1) and has
much profited from comments of the reviewers.
Appendix A. Regimes and model parameters
For all applied rheologies we have identified different dynamical regimes, each characterised by a specific behaviour of the surface region. These domains
exhibit (a) a stagnant lid, (b) a mobile, fluid-like surface and (c) a plate-like surface (see Section 5). Similar domains have been found in 2D flows (Moresi and
Solomatov, 1998).
It is useful to map out these domains, since it allows
to extrapolate the effect of varying parameters (viscosity contrast, Rayleigh number, etc.). Many model
runs were necessary in order to do so and to localise
the transitions between these regimes. The runs were
performed at lower resolution (32 × 32 × 16) and
serve to roughly distinguish the evolutionary paths,
rather than determine the transitions with a high
precision.
The following tables list all data on which the domain diagrams are based. All input parameters and the
resulting states are shown.
250
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
Table 3 Data set of runs used to create Fig. 2a
Table 5 Data set of runs used to create Fig. 10
Varied parameters
Resulting
regime
Varied parameters
Constant yield
stress, σ 0
Depth-dependent
yield stress, σ z
Mobile
Stagnant
Mobile
Stagnant
Mobile
Episodic
Stagnant
Mobile
Episodic
Episodic
Stagnant
Mobile
Episodic
Episodic
Stagnant
Constant parameters: Ra = 200, RD = 20, σz = 0,
∆α = 1, Q = 0.
0.001
0.001
0.001
0.1
0.1
0.1
0.1
1
1
1
4
4
4
4
7
7
7
10
120
130
150
100
120
130
150
50
100
120
10
30
80
100
0.5
50
80
0.001
Table 4 Data set of runs used to create Fig. 2b
Constant parameters: η = 105 , Ra = 200, RD =
20, α = 1, Q = 0.
Viscosity
contrast, η
Yield stress,
σ0
104
104
5 × 104
5 × 104
105
105
105
5 × 105
5 × 105
5 × 105
5 × 105
106
106
106
106
8
9
7
8
6.5
7
8
5.5
6
7.5
7.8
5
5.5
7
7.5
Varied parameters
Rayleigh
number, Ra
Yield
stress, σ 0
100
100
150
150
150
200
200
200
250
250
250
250
300
300
5
6
6
6.5
7
6.5
7
8
8
8.5
12
13
9
10
Resulting
regime
Constant parameters: η = 105 , RD = 20, σz = 0,
α = 1, Q = 0.
Mobile
Episodic
Stagnant
Mobile
Episodic
Episodic
Stagnant
Mobile
Episodic
Stagnant
Mobile
Episodic
Episodic
stagnant
Episodic
Episodic
Stagnant
Stagnant
Table 6 Data set of runs used to create Fig. 12
Varied parameters
Mobile
Stagnant
Mobile
Episodic
Stagnant
Mobile
Episodic
Stagnant
Mobile
Episodic
Episodic
Stagnant
Mobile
Episodic
Resulting
regime
Depth-dependent
viscosity, RD
1
1
1
1
5
5
5
5
10
10
10
10
10
10
Yield stress,
σ0
1
2
4
7
2
3
6.5
7
1.5
2
3
4
7
7.5
Resulting
regime
Mobile
Episodic
Episodic
Stagnant
Mobile
Episodic
Episodic
Stagnant
Mobile
Plate
Plate
Episodic
Episodic
Stagnant
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
Appendix A (Continued )
Varied parameters
Depth-dependent
viscosity, RD
Yield stress,
σ0
20
20
20
20
20
30
30
30
30
100
100
100
100
2
3
6.5
7
8
1
3
8
9
1
5
9
11
Appendix A (Continued )
Resulting
regime
Varied parameters
Internal
heating, Q
Yield stress,
σ0
Mobile
Plate
Plate
Episodic
Stagnant
Mobile
Plate
Plate
Stagnant
Mobile
Plate
Plate
Stagnant
5
10
10
4
0.5
1
Constant parameters: η = 105 , Ra = 200, σz =
0, α = 1, Q = 0.
Table 7 Data set of runs used to create Fig. 18
Varied parameters
Internal
heating, Q
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
5
5
5
Yield stress,
σ0
2
4
6.5
7
8
1
3
4
5
6
7
1
3
4
6.5
6.8
1
1.5
2
251
Resulting
regime
Mobile
Plate
Plate
Episodic
Stagnant
Mobile
Plate
Plate
Episodic
Episodic
Stagnant
Mobile
Plate
Episodic
Episodic
Stagnant
Mobile
Plate
Episodic
Resulting
regime
Stagnant
Mobile
Stagnant
Constant parameters: η = 105 , Ra = 200, RD =
20, σz = 0, α = 1.
Table 8 Data set of runs used to create Fig. 23
Varied parameters
Depth-dependent
expansivity, 1/α
Yield stress,
σ0
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
5
5
5
7
7
7
2
3
6.5
7
8
1
2
6
7
7.5
1
1.5
5
6
7
1
4
5
2
4
4.5
Resulting
regime
Mobile
Plate
Plate
Episodic
Stagnant
Mobile
Plate
Plate
Episodic
Stagnant
Mobile
Plate
Plate
Episodic
Stagnant
Plate
Plate
Stagnant
Plate
Plate
Stagnant
Constant parameters: η = 105 , Ra = 200, RD =
20, σz = 0, Q = 0.
The variation of the Nusselt number with time
serves as a criterion to distinguish between the
regimes. Fig. 28 illustrates the temporal evolution of
the Nusselt number for the four regimes. In the stagnant lid regime (thick solid line) the Nusselt number
252
C. Stein et al. / Physics of the Earth and Planetary Interiors 142 (2004) 225–255
100
80
Nu
60
mobile
episodic
40
stagnant
plate
20
1
0.06
0.08
0.1
0.12
0.14
0.16
0.18
time
Fig. 28. Illustration of the difference in behaviour of the temporal
evolution of the Nusselt number for the four convective regimes.
is low due to the purely conductive transport in the
immobile surface. In the mobile lid regime (thin solid
line) the Nusselt number is higher and strongly timedependent. The episodic regime (dashed line) shows
intermittent bursts. In the plate regime (dotted line)
the episodic behaviour fades away and a steady state
is obtained.
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