Arrhenius rheology versus Frank-Kamenetskii rheology—Implications for mantle dynamics



[1] The viscosity of planetary mantle material is strongly temperature dependent. This dependence is described by an Arrhenius law. But for the realistic viscosity contrast that appears over the depth of the mantle, strong gradients in the upper thermal boundary layer occur. These strong gradients are not realizable in numerical models. Therefore, the Frank-Kamenetskii approximation, leading to a linearized exponential viscosity function, is commonly used. Much research on the plate-mantle system has been done applying the Frank-Kamenetskii rheology. The question though still arises, if these results can be reproduced when using the Arrhenius law. Here it has to be kept in mind, that in numerical models the realistic viscosity contrast appearing over the mantle can neither be coped with by the Arrhenius nor the Frank-Kamenetskii approach. Thus, the computational aspects to date only allow for viscosity contrasts being moderate as compared to realistic values and extrapolations to planet-like values have to be made for both rheologies. Comparing results obtained by the commonly used forms of the Arrhenius and the Frank-Kamenetskii approach, we observe the same change in flow behavior from mobile-lid to stagnant-lid convection. The differences are only of quantitative nature: In the stagnant-lid regime, some Arrhenius formulations lead to a thinner top boundary layer which results in values of lithospheric thicknesses being more realistic. Here it has to be noted that different forms of the Arrhenius law have been used which differ among themselves. When properly scaled, the differences between the Frank-Kamenetskii and Arrhenius rheology can, however, be strongly diminished. To understand general features in the plate tectonics-mantle convection system an additional stress dependence of the viscosity has to be considered. The discrepancies between the various viscosity formulations are then of even less importance, because for an Earth-like convection regime, the top viscosity is more strongly influenced by the stress dependence rather than by the temperature dependence.

1. Introduction

[2] Viscosity, varying with temperature and stress, is known to strongly influence convective dynamics in the mantle of terrestrial planets. The cold stagnant-lid forming atop the convection cell can be deformed, enabling plate-like mobility of the surface [e.g., Solomatov, 1995; Trompert and Hansen, 1998b; Moresi and Solomatov, 1998].

[3] The basis for the formation of a plate is the temperature dependence of the viscosity. Convection with temperature-dependent viscosity (also named thermoviscous convection) has been widely investigated analytically and by laboratory experiments [e.g., Nataf and Richter, 1982; Morris and Canright, 1984; Fowler, 1985; Davaille and Jaupart, 1993], by numerical experiments in a 2-D Cartesian geometry [e.g., Christensen, 1984a, 1984b; Moresi and Solomatov, 1995; Chen and King, 1998; Dumoulin et al., 1999], in a 3-D Cartesian geometry [e.g., Tackley, 1993; Balachandar et al., 1995; Christensen and Harder, 2007], and in a spherical geometry [e.g., Ratcliff et al., 1996; Stemmer et al., 2006; Zhong et al., 2008]. In a theoretical analysis, Solomatov [1995] classified three flow regimes in thermoviscous convection. First, for a weak temperature dependence, the surface viscosity is not high enough to prevent the movement of the top layer. The surface velocity is comparable to the interior velocity, so that this small viscosity contrast regime is still similar to isoviscous convection. Second, in the transitional regime occurring for an intermediate temperature dependence, a Rayleigh number-dependent boundary exists between the small viscosity contrast regime and the stagnant-lid regime [Trompert and Hansen, 1998a]. The latter is the third flow regime occurring for a strong temperature dependence. This leads to such high viscosities in the cold material at the top, that surface velocity declines to zero. A stagnant lid covers the complete surface and convection is confined to the area beneath this lid [e.g., Booker, 1976; Ogawa et al., 1991; Ratcliff et al., 1996, 1997; Grasset and Parmentier, 1998; Reese et al., 1999a, 2005].

[4] Besides the temperature dependence, the viscosity of mantle material is also stress dependent [Karato and Wu, 1993]. A similar classification in different flow regimes was observed for temperature- and stress-dependent viscosity convection [e.g., Moresi and Solomatov, 1998; Tackley, 2000a, 2000b; Stein et al., 2004]. Using a yield stress, the system can either be dominated by the temperature dependence, stress dependence or both parts are in balance. For a strong temperature dependence, the system thus changes from the stagnant-lid regime to an episodically mobilized and further to the mobile-lid regime with increasing stress dependence [e.g., Stein and Hansen, 2008]. In the mobile-lid regime, the surface is strongly weakend due to high stresses and convection resembles isoviscous convection [Tackley, 2000a; Stein et al., 2004]. Thus, this regime is comparable to the small viscosity contrast regime in thermoviscous convection. For an intermediate stress and a strong temperature dependence of the viscosity, on short timescales, rigidly moving surface pieces (i.e., plate-like structures) are found [Trompert and Hansen, 1998b].

[5] Mantle rheology is poorly constrained. One method of determining mantle properties are deformation experiments made on olivine (the most abundant material in the upper mantle) conducted under laboratory conditions. Here however, the rate of deformation is much faster than in Earth, so that stresses are much higher in laboratory experiments [Karato, 2003]. Additionally, the temperatures are higher whereas pressures are too low requiring extrapolations of laboratory results over many orders of magnitude [Schubert et al., 2001; Karato, 2003]. The resulting dislocation creep (non-Newtonian power-law rheology) found in laboratory experiments seems inconsistent with the diffusion creep (Newtonian viscosity) favored by studies of postglacial rebound. Most likely deformation is a function of pressure with dislocation creep as the dominant mechanism in the shallow upper mantle and diffusion creep in the deep upper mantle [Karato and Wu, 1993]. The effect of a pressure-dependent viscosity in mantle convection was shown to be of great importance [e.g., Stein and Hansen, 2008]. As the exact form of the stress dependence is not known, different forms have been considered [e.g., Christensen, 1983; Weinstein and Olson, 1992]. Bercovici [1993] finds that a nonlinear viscosity with a stress exponent larger than 11 is required in order to achieve plate-like behavior, but Schubert et al. [2001] suggest that a creep process according to a power-law might not be the only mechanism valid in the lithosphere because deformation experiments with mantle material only delivered an exponent of 3 ≤ n ≤ 5. Studies that combine the temperature dependence with different forms of Newtonian rheology all reveal the same classifications in flow regimes [Moresi and Solomatov, 1998; Tackley, 2000a; Stein et al., 2004].

[6] Regarding the temperature dependence, deformation experiments of mantle material are fit by an Arrhenius-typed law [cf., Schubert et al., 2001]. In numerical models, however, the extreme gradients arising in the top layer lead to problems. Accordingly, the law is often simplified by using an approximation in form of a linearized exponential function [Frank-Kamenetskii, 1969].

[7] Moresi and Solomatov [1995] suggest that the linearized approximation is valid for high-viscosity contrasts because in the stagnant-lid regime convection is driven by a rheological temperature scale rather than by the total viscosity contrast. Several discussions on the validity of the approximation were given [Morris, 1982; Morris and Canright, 1984; Ansari and Morris, 1985; Fowler, 1985; Ratcliff et al., 1997] but direct comparisons are rare. Reese et al. [1999b] provide one example from the stagnant-lid regime. A similar example was used in Solomatov [2004] to discuss the effect of the viscosity on the stress distribution in the lid. King [2009] showed that it is possible to find a scaling between the Arrhenius rheology and the Frank-Kamenetskii approximation in which the Nusselt number and root-mean-square velocity fit, but the topography differs significantly. In addition to the different Frank-Kamenetskii approximations, various forms of the Arrhenius rheology have been used [e.g., Huang et al., 2003; Korenaga, 2005; King, 2009].

[8] A comparison of all approximations has not yet been given. Therefore, the aim of our work is to contrast the commonly applied approximations against each other using the same parameter conditions. In our study we additionally extend the comparison between the Arrhenius rheology and the Frank-Kamenetskii approximation by examining purely thermoviscous convection for a wide range of viscosity contrasts. First, we present the parameter space spanned by the system parameters (Rayleigh number and viscosity contrast) to investigate the general flow behavior. Then we show results from the stagnant-lid convection for a strong temperature dependence, which we compare to a case of the mobile-lid convection for an additionally strong stress dependence of the viscosity. Thus, we also provide a discussion on the influence of the stress dependence which plays an important role when considering plate tectonics. We, however, neglect further aspects that are important for plate modeling (e.g., water, pressure dependence, grain-size dependence, melting) to keep the comparison as simple as possible.

[9] We instead focus on presenting and comparing results for different types of Arrhenius approaches that are commonly used in numerical experiments [Solomatov and Moresi, 2002; Korenaga and Jordan, 2003; Vezolainen et al., 2003, 2004; Korenaga, 2005] and for an adjusted Frank-Kamenetskii rheology [King, 2009]. In doing this, we suggest that the use of the Frank-Kamenetskii approximation is as good as the Arrhenius rheology due to the improper knowledge of material parameters in the planets' interiors.

2. Models

[10] The governing equations for mantle convection are deduced from the conservation of mass, energy and momentum. With the assumption of an incompressible fluid and the Boussinesq approximation the nondimensional equations are:

display math(1)
display math(2)
display math(3)

where inline image, T, p, and inline image are the velocity vector, the temperature, the pressure and the unit vector in the vertical direction z, respectively.

[11] The nondimensionalization of equations (1)-(3) has been achieved by using the layer thickness d, the thermal diffusion time inline image, the surface viscosity inline image and the temperature drop inline image to scale the length, the time and the temperature, respectively.

[12] This scaling leads to the Rayleigh number:

display math(4)

where α, ρ, g, inline image, d, κ and η are the thermal expansion coefficient, the density, the gravitational acceleration, the temperature difference across the layer, the thickness of the layer, the thermal diffusivity and the viscosity, respectively. As the viscosity varies spatially, one has to carefully define what viscosity is used in the definition of the Rayleigh number. In this study, the Rayleigh number is evaluated using the viscosity at the top. Besides this definition of the Rayleigh number others are known [Schubert et al., 2001], but the advantage of the top Rayleigh number is that it is a priori known.

[13] The equations are solved with a finite volume multigrid method [cf., Trompert and Hansen, 1996] in a two-dimensional Cartesian geometry. A square box with a resolution of 642 control volumes is used (for the high Rayleigh numbers used for the regime plot in Figure 2 resolutions up to 2562 have been used). Additionally a grid refinement is applied to resolve the strong gradients appearing in the thermal boundary layers.

[14] Stress-free boundary conditions for the velocities and a vanishing heat flux for the temperature at the sidewalls is assumed. The temperature at the top and bottom boundaries are set to T = 0 and T = 1, respectively. Two types of initial conditions for the temperature have been used: inline image and inline image. The latter was chosen because T = 0.92 resembles the internal temperature in stagnant-lid convection [Solomatov, 1995]. No qualitative difference between the constantly hot and the conductive starting field or the different kinds of perturbations could be observed during modeled time. For the discussion of the time-dependent, high Rayleigh number cases we use the ensemble-average (i.e., we average over time in both model runs) to minimize the effect of possible different evolutionary branches.

2.1. Viscosity

[15] The effective viscosity in this study is given as harmonic mean of the temperature-dependent component inline image and the stress-dependent component inline image:

display math(5)

[16] The stress dependence is applied in form of a Bingham model with a yield criterion [Trompert and Hansen, 1998b] as:

display math(6)

which was shown to result in the same flow behavior as the rheologies applied in Moresi and Solomatov [1998] and Tackley [2000a]. In this study, the plastic viscosity inline image is kept constant to 0.001. Lower values of the plastic viscosity yield a fluid-like surface, higher values result in a stagnant lid that cannot be deformed. inline image is the yield stress and ϵ the second invariant of the strain rate tensor.

[17] The temperature dependence of the viscosity is expressed by the Arrhenius law:

display math(7)

where C is the normalized pre-exponential viscosity, E is the activation energy, R the gas constant, and inline image is the dimensional temperature.

[18] Using the temperature drop inline image to scale the temperature and scaling the activation energy as inline image leads to the nondimensional form of the Arrhenius rheology:

display math(8)

with the constant inline image, the activation temperature A and inline image the nondimensional surface temperature or offset temperature. The tilde marks the dimensional quantity.

[19] This law is approximately expressed by a linearized form [Frank-Kamenetskii, 1969] as:

display math(9)

inline image and inline image are the reference viscosity and temperature, respectively. In our model we use the top values inline image and inline image for defining the Rayleigh number. Independent of this choice, the viscosity contrast is [cf., Schubert et al., 2001]:

display math(10)

[20] We here refer to the linearized temperature dependence, that is widely used in numerical studies [e.g., Christensen, 1984b; Solomatov, 1995; King, 2009; Solomatov, 2012], as Frank-Kamenetskii rheology.

[21] For the comparison of both formulations (equations (8) and (9)) the viscosity structures must resemble each other in some way. There are many ways to achieve this, e.g., by the normalization to either the same top and bottom viscosity or the same interior viscosity. While the latter assumption should give better results as the viscosity in the actually convecting part is the same, the first assumption is more widely used as the viscosities at the boundaries are a priori known. The normalization is controlled by the prefactor b in the Arrhenius law (equation (8)). In our study, we present some of the typically used formulations of the Arrhenius rheology and compare their results.

[22] For the normalization to the top viscosity, the Arrhenius law (equation (8)) can be rewritten as:

display math(11)

[23] This formulation is similar to the ones used in Solomatov and Moresi [2002], Korenaga [2005], and Rolf and Tackley [2011] and will in the following be simply referred to as Arrhenius rheology.

[24] Applying further the condition that the viscosity contrast for both rheologies is equal, we can derive the following relation between the activation temperature A and the offset temperature T0:

display math(12)

[25] Realistic values for the activation and offset temperature can be estimated from realistic values of the activation energy E, the gas constant R, the surface temperature inline image and the temperature drop inline image over the mantle. The value for the gas constant is R = 8.3144 J K−1 mol−1. For diffusion or dislocation creep in a dry upper mantle, the activation energy, E, is 300–540 kJ mol−1 [Karato and Wu, 1993; Kirby, 1983]. The surface temperature is about 300 K and the temperature at the core-mantle boundary about 3750–4150 K [van der Hilst et al., 2007]. Altogether this gives: A ≈ 9–17 and for the offset temperature T0 ≈ 0.07–0.08 which yields a realistic viscosity contrast of the order of inline image and larger. Modeling of high-viscosity contrasts is, however, strongly limited by the computational power. Today a viscosity contrast of about inline image is feasible in numerical models of mantle convection. Here it has to be kept in mind that deformation is present besides the strong temperature dependence which would also reduce the temperature-dependent viscosity contrast to more moderate values.

[26] Using the same interior viscosities for normalization should result in a similar solution at very high viscosity contrasts, because the actual convection takes place in the approximately isothermal interior [Solomatov and Moresi, 1996]. Therefore, we also consider a normalization where we equate the interior viscosities of the Arrhenius rheology (equation (8)) and the Frank-Kamenetskii approximation:

display math

[27] This gives the prefactor

display math(13)

and the interior-Arrhenius rheology

display math(14)

[28] For this Arrhenius rheology, the difficulty, however, is that the interior temperature Ti is not a priori known. Here we follow the method of Moresi and Solomatov [1995] to compute the interior temperature.

[29] Another commonly applied Arrhenius law is a formulation with an imposed cutoff viscosity [e.g., Huang et al., 2003; Huang and Zhong, 2005; King, 2009]. In the cutoff-Arrhenius rheology:

display math(15)

the bottom viscosity is equal to that of the Frank-Kamenetskii rheology. The purpose of this formulation is, as in the Frank-Kamenetskii rheology, to limit the strong gradients appearing in the top boundary layer, where the velocities are already nearly zero. This is done by cutting too large viscosity values off and setting the topmost viscosity to a constant value instead inline image. As shown in King [2009] we also find that the results of this rheology are unchanged as long as the viscosity cutoff, inline image, is at least 104 times the bottom viscosity.

[30] Figure 1a shows the Arrhenius and Frank-Kamenetskii viscosities as function of the temperature which varies between 0 at the surface and 1 at the bottom of the model domain. The viscosity contrast here is inline image. We observe that for the Frank-Kamenetskii rheology the viscosity gradient in the convection region is different to that of the other rheologies. Solomatov and Moresi [1996] argue, however, that the viscosity gradients in this region should be equal to give similar results.

Figure 1.

(a) Frank-Kamenetskii viscosity inline image (equation (9)), Arrhenius viscosity inline image (equation (11) with inline image and A = 1.52), interior-Arrhenius viscosity inline image (equation (14) with inline image, A = 1.52 and inline image), cutoff-Arrhenius viscosity inline image (equation (15) with inline image and A = 6.33), derivative-Arrhenius viscosity inline image (equation (16) with inline image) and the adjusted Frank-Kamenetskii viscosity inline image (equation (17) with inline image and inline image). (b) Frank-Kamenetskii viscosity and Arrhenius viscosity (equation (11)) with offset temperatures ranging from T0 = 0.01–10. The viscosity contrast for all functions is inline image.

Figure 2.

Regime diagram spanned by the viscosity ratio inline image and the bottom Rayleigh number inline image normalized by the critical bottom Rayleigh number for the Frank-Kamenetskii rheology and the Arrhenius rheology (equation (11)). The dots represent the modeled results and the lines mark the regime transitions for the Frank-Kamenetskii rheology adapted from Solomatov [1995].

[31] Therefore, besides the formulations in which we equate the viscosity and the viscosity contrast (equations (11), (14), and (15)), we also used a conversion method in which instead of the viscosity contrast the temperature derivative of the viscosity is the same. Thus, equating the viscosities plus the gradient of the viscosity at the bottom temperature leads to:

display math(16)

[32] This approach is very similar to the one we use in the cutoff-Arrhenius viscosity (equation (15)) but the viscosity variations are larger in the derivative-Arrhenius approach so that the viscosity is truncated over a larger depth range (cf., Figure 1a).

[33] Finally, in Figure 1a we also show a second Frank-Kamenetskii formulation:

display math(17)

[34] Following King [2009] we slightly deviated from the conversion method by Solomatov and Moresi [1996] and obtained β and Ti by trial and error so that the Nusselt number and the root-mean-square velocity in this adjusted Frank-Kamenetskii formulation match those in the Arrhenius formulation inline image. For a viscosity contrast of inline image, we choose inline image and inline image.

[35] In Figure 1b, for the Arrhenius function inline image (equation (11)), we show examples with the same viscosity contrast but varying offset temperatures (T0 = 0.01, 0.1, 1, and 10). The activation temperature A is adjusted according to equation (12) and is inline image, inline image, A1 = 27.64 and A10 = 1520.2. The larger the offset temperature (and the activation energy), the more the Arrhenius viscosity and the Frank-Kamenetskii viscosity are alike. For a realistic activation energy, the offset temperature is too low. On the other hand, T0 = 0.1 is a realistic offset temperature, but here the activation energy is too low. Thus, both assumptions are not fully realistic, which is due to the unrealistically low viscosity contrast.

[36] In this study, we have decided on an offset temperature of inline image for all Arrhenius formulations. For the cutoff viscosity, the activation energy is A = 6.33. This would result in a viscosity contrast of 1025, but values of the viscosity larger than inline image are set to inline image.

[37] Table 1 summarizes the viscosity formulations that have been discussed in this section and that will be used in the following.

Table 1. Summary of all Rheological Formulations Useda
  1. a

    inline image is the activation temperature, inline image the viscosity ratio, T0 the offset temperature, Ti the interior temperature, and β an adjusted viscosity ratio. All values are nondimensional quantities.

Frank-Kamenetskii approximation inline image
Arrhenius law inline image
Interior-Arrhenius law inline image
Cutoff-Arrhenius law inline image
Derivative-Arrhenius law inline image
Adjusted Frank-Kamenetskii law inline image

3. Numerical Experiments

[38] Following the work of Solomatov [1995], we first explore the parameter range spanned by the bottom Rayleigh number inline image (normalized by the critical bottom Rayleigh number) and the viscosity contrast inline image. This regime diagram allows for an extrapolation to planet-like parameters, i.e., higher Rayleigh numbers and viscosity contrasts than can actually be achieved in experiments.

[39] Figure 2a shows the regime diagram for the Frank-Kamenetskii rheology and Figure 2b for the Arrhenius rheology (equation (11)) with inline image [the function differing the most from the Frank-Kamenetskii function (cf., Figure 1a)]. The lines in both plots mark regime transitions obtained for the Frank-Kamenetskii rheology according to Solomatov [1995] and the points represent our results. The regimes have been identified by means of differences in the flow behavior. Quantitative measures are the mobility M (which is the ratio of surface to total root-mean-square velocity) and the interior temperature. The diamonds stand for runs in which the Rayleigh number is subcritical and no convection appears. The filled circles represent mobile-lid convection (i.e., the small viscosity contrast regime). Here the surface velocity is comparable to the interior velocity which gives a mobility of M > 0.9 and the internal temperature is approximately 0.5. In stagnant-lid convection (marked with squares) the mobility is strongly reduced (M < 0.1). Sluggish-lid convection, as obtained in the transitional regime (indicated by open circles), is characterized by a mobility 0.9 ≥ M ≥ 0.1 and an internal temperature larger than 0.5. Applying these conditions, our Frank-Kamenetskii results largely support the regime transitions as reported by Solomatov [1995]. We only observe that the transition to stagnant-lid convection is not at a constant viscosity contrast but is slightly dependent on the Rayleigh number. In addition, for some cases with a low viscosity contrast and high Rayleigh number we find that the system changes back from mobile-lid convection to sluggish-lid convection. Here the mobility is close to the critical transition value of 0.9. For the Arrhenius rheology we also observe the three convection regimes. Especially, we also find that at low viscosity contrasts and high Rayleigh numbers the systems fluctuate around the critical transition value of 0.9 and that the transition to stagnant-lid convection is Rayleigh number dependent. However, the transition to stagnant-lid convection is shifted to slightly lower parameter values which can be explained by the viscosity structure.

[40] The change in the flow pattern occurring with an increasing viscosity contrast is shown in Figure 3 by means of the snapshots of the temperature field. For all rheologies, we clearly observe the change from isoviscous-like convection to stagnant-lid convection as the viscosity contrast increases. For the high viscosity contrasts the topmost layer does no longer participate in the convective cycle and the interior heats up. As with increasing viscosity contrast, the Rayleigh number is locally increased, a thinning of the rising and sinking structures is observed. As expected from Figure 1a we find a good agreement between the Frank-Kamenetskii rheology inline image (Figure 3a) and the derivative-Arrhenius viscosity inline image (Figure 3d) as well as between the adjusted Frank-Kamenetskii rheology inline image (Figure 3b) and the Arrhenius rheology inline image (Figure 3c). The latter two systems are more strongly supercritical explaining the observance of thinner plumes and boundary layers for the systems in Figures 3b and 3c.

Figure 3.

Snapshots of the temperature fields for model calculations with a (top) Rayleigh number of Ra = 100 and viscosity contrasts ranging from inline image. The snapshots were taken at a characteristic time in the (statistically) steady state, except for the viscosity contrast of 106 where the snapshots represent the ensemble-averaged temperature field. The two left columns represent the Frank-Kamenetskii rheologies (a) inline image and (b) inline image, the two right columns the Arrhenius rheologies (c) inline image and (d) inline image (both with inline image). (The parameters for the adjusted Frank-Kamenetskii are: inline image 6.0 and Ti = 0.71 for inline image, inline image and Ti = 0.74 for inline image, inline image and Ti = 0.78 for inline image, inline image and Ti = 0.83 for inline image, inline image and Ti = 0.85 for inline image).

[41] Figure 4 shows the relationship between the Nusselt number and the Rayleigh number for viscosity contrasts ranging between inline image and 106. (Note that here we are not showing the results of the adjusted Frank-Kamenetskii law as this formulation has been chosen to match the Nusselt number of the Arrhenius law.) The lines indicate the fits to the calculated results which are marked as symbols. For the Frank-Kamenetskii rheology we find slopes in the fits ranging from 0.31 (for the viscosity contrast of inline image) to 0.29 (for the higher viscosity contrasts). For the Arrhenius rheology the slopes decrease in a similar way from 0.28 to 0.27 with increasing viscosity contrasts. Again, the derivative-Arrhenius results (slopes ranging from 0.32 to 0.29) resemble more the Frank-Kamenetskii results than the Arrhenius results. In general, the values are comparable to previously observed results [Christensen, 1984b]. Also the reduction in the slope with increasing viscosity contrast has already been observed for the Frank-Kamenetskii approximation [Christensen, 1984b; Reese et al., 1998].

Figure 4.

Nusselt number-Rayleigh number relationship for (a) the Frank-Kamenetskii rheology, (b) the Arrhenius rheology and (c) the derivative-Arrhenius rheology. Both Arrhenius formulations use inline image. The symbols mark the time-averaged values of the calculations and the lines the fits.

[42] To summarize, we observe the same qualitative behavior for the Frank-Kamenetskii approximation and the Arrhenius rheology. When scaled correctly, also quantitative results of both rheologies agree well. We will now extend the comparison to different types of the Arrhenius rheology commonly applied. Here we will limit the study to two extreme cases: the stagnant-lid regime (in thermoviscous convection), which presumably is most relevant for the terrestrial planets, and to the mobile-lid convection appearing for an additional stress dependence. This has explained plate-like behavior very well [e.g., Moresi and Solomatov, 1998; Trompert and Hansen, 1998b].

3.1. Stagnant-Lid Convection (Thermoviscous Convection)

[43] Figure 5 shows snapshots of the temperature fields for the different viscosity formulations (cf., Figure 1a). All snapshots clearly display the stagnant lid, i.e., the coldest layer does not develop any downwellings. Small instabilities instead detach from the rheological sublayer located beneath the stagnant lid [Solomatov and Moresi, 2000] (not visible in the ensemble-averaged temperature fields). Again, the obvious difference between the temperature fields is the thinner stagnant lid and the colder bulk for some Arrhenius rheologies [cf., King, 2009]. Here however, a significant difference among the different Arrhenius approaches can be observed. As expected from Figure 1, the Arrhenius rheology, the interior-Arrhenius rheology and less prominent the adjusted Frank-Kamenetskii rheology differ most strongly from the Frank-Kamenetskii rheology, while the cutoff-Arrhenius and derivative-Arrhenius rheology most closely resemble it. This is also evident in the corresponding depth profiles displayed in Figure 6. We observe that all Arrhenius rheologies show a stronger gradient in the top viscosity than the two Frank-Kamenetskii viscosities (Figure 6a), but the temperature profiles for the Arrhenius, the interior Arrhenius and the adjusted Frank-Kamenetskii rheology are almost identical (Figure 6b). They show a thinner top boundary layer and a lower interior temperature than the three other profiles that are also very alike. The lowest interior temperature (Ti ≈ 0.8) appears for the Arrhenius rheology with inline image and the highest for the derivative-Arrhenius rheology (Ti ≈ 0.94). The velocity-depth profiles (Figure 6c) all show the strong velocity decrease over the top boundary and an almost constant interior velocity but with strong differences in the velocity values. Figure 6d shows the surface topography for the different viscosity formulations. For the Arrhenius rheologies we find several undulations in the surface topography as small-scale convection is present. Additionally we find that the Arrhenius rheologies and the Frank-Kamenetskii rheology have a much smaller amplitude in topography than the adjusted Frank-Kamenetskii rheology. A significant difference in topography between the Arrhenius and Frank-Kamenetskii rheology was also found by King [2009], even the adjusted Frank-Kamenetskii formulation (which has been chosen in such a way that Nusselt numbers and rms-velocities were in accord with those of the Arrhenius rheology) gives a different topography with a much higher amplitude. The differences can be explained by the flow pattern. In the Arrhenius rheologies, the critical Rayleigh number is lower so that at Ra=100 the flow is strongly time-dependent and only leads to small instabilities. These cause a small amplitude topography [cf., Stein et al., 2010]. In the Frank-Kamenetskii rheologies the critical Rayleigh number is slightly higher and the flow is still largely dominated by a cell structure. Thus, the topography in these rheologies show less undulations but rather reflect the strong plume in the middle of the box. Additionally, the thicker stagnant lid in the Frank-Kamenetskii rheology reduces the amplitude [cf., Stein et al., 2010] compared to the case with a thinner lid for the adjusted Frank-Kamenetskii rheology.

Figure 5.

Snapshots of the temperature fields (ensemble-averaged) for model calculations with a viscosity contrast of inline image and a Rayleigh number of Ra = 100. The snapshots represent stagnant-lid convection for (a) the Frank-Kamenetskii rheology, (b) the Arrhenius rheology with inline image and A = 1.52, (c) the interior-Arrhenius rheology (bi = 14.52, inline image and A = 1.52), (d) the cutoff-Arrhenius viscosity, (e) the derivative-Arrhenius viscosity with inline image, and (f) the adjusted Frank-Kamenetskii viscosity with β = 9.1 and Ti = 0.85.

Figure 6.

(a) Viscosity, (b) temperature, (c) velocity, and (d) topography profiles (ensemble-averaged) for the model calculations presented in Figure 5.

[44] The general observation from this plot is that all approaches lead to the stagnant-lid configuration for inline image and Ra = 100. The choice of rheological parameters and the type of viscosity law controls the output parameters, which partly deviate strongly (also among the different Arrhenius laws). These deviations can be diminished when properly scaled. Again we observe that the cutoff-Arrhenius and derivative-Arrhenius rheology agree well with the Frank-Kamenetskii rheology.

3.2. Mobile Lid Convection

[45] We also investigated the influence of additional stress dependence on the flow. The three flow regimes described in Moresi and Solomatov [1998] are observed for all approaches, where again a shift in the regime transition for the Arrhenius rheology is obtained. We will here only present results of the mobile-lid regime that appears for a strong stress dependence. A weaker dependence of the stress leads to the already discussed stagnant-lid convection or an intermittent behavior, that temporally changes between the mobile and the stagnant-lid configuration.

[46] Figure 7 displays snapshots of the temperature fields in the mobile-lid regime. We find that the coldest material now takes part in the convective cycle and thus the interior is effectively cooled compared to the stagnant-lid convection. Figure 8a shows the reason for this: the viscosity-depth profiles for all approaches display that the viscosity drop over the top boundary is reduced compared to the stagnant-lid case, reducing the effective viscosity contrast. Thus, the stagnant lid breaks and the interior is cooled, which can also be seen in the temperature-depth profiles in Figure 8b, where interior temperatures vary between 0.59 and 0.75. The velocity does no longer decrease strongly over the top boundary (Figure 8c) but the surface velocity is almost comparable to the interior velocity. For this mode of convection, the different profiles vary strongly among the different rheologies which again will be due to the different critical Rayleigh numbers. Interestingly, in this regime, the deviations in topography (Figure 8d) among all viscosity formulations are reduced because the different temperature-dependent viscosity structures play a minor role in the overall viscosity compared to the stress-dependent viscosity [cf., Stein and Hansen, 2008]. In all cases the topography reflects the plume on the left side, where the topography is elevated, and the subduction zone on the right side which leads to a strong depression. Slightly deviating is the topography in the adjusted Frank-Kamenetskii rheology as we do not observe a steady one-cell convection structure but rather the upwelling is moving from the middle to the left box boundary where it is competing with the downwelling.

Figure 7.

Snapshots of the temperature fields (ensemble-averaged) for model calculations in the mobile-lid regime (viscosity contrast of inline image, Rayleigh number of Ra = 100, yield stress of inline image). (a) The Frank-Kamenetskii rheology, (b) The Arrhenius rheology with inline image, (c) The interior-Arrhenius rheology (bi = 10.62, inline image), (d) The cutoff-Arrhenius rheology with A = 6.33 and inline image, (e) The derivative-Arrhenius viscosity with inline image, and (f) The adjusted Frank-Kamenetskii viscosity with β = 8.8 and Ti = 0.83.

Figure 8.

(a) Viscosity, (b) temperature, (c) velocity, and (d) topography profiles (ensemble-averaged) for the model calculations presented in Figure 7.

[47] In the mobile-lid regime, the surface viscosity is dominated by the stress dependence [Stein and Hansen, 2008]. This leads to a strong reduction of the surface viscosity, minimizing the effect of the temperature-dependent viscosity component. Solomatov [2004] also showed that, in the presence of a yield stress, the stress profiles for the Arrhenius and Frank-Kamenetskii rheology are not much different, because the dependence on the plastic zone is stronger than on the viscosity contrast due to temperature.

4. Discussion

[48] As there still are concerns with the linearized temperature-dependent viscosity in mantle convection models, we investigated if the Frank-Kamenetskii rheology is a valid approximation for the Arrhenius-typed temperature dependence of the viscosity in a planet's mantle. Additionally, we addressed various forms of Arrhenius laws and have shown that the general observation (i.e., the formation of a stagnant-lid at high viscosity contrasts) is the same for all commonly used viscosity laws.

[49] The quantitative values among all rheologies vary, but as we still only have imprecise knowledge of realistic mantle parameters, it has to be kept in mind that a “correct” viscosity law is not precisely known. We have shown that the viscosity contrast between the surface and the core-mantle boundary for the Arrhenius rheology is determined by both the activation energy and the offset temperature. Using somewhat realistic values for both parameters leads to viscosity contrasts of the order of 1050 and more, which is not feasible in numerical models today. Therefore reduced viscosity contrasts have to be considered, where either of the parameter is chosen to be in the right balance or viscosity is cutoff. We have presented viscosity profiles for the cases: (i) a realistic offset temperature but an unrealistic activation temperature and (ii) an unrealistic offset temperature but a realistic activation temperature. For the same viscosity contrast, both assumptions lead to quite different viscosity structures with the latter case being very similar to the Frank-Kamenetskii viscosity.

[50] We generally observe that the scaling is very important. When properly scaled we find a good agreement between the Frank-Kamenetskii and Arrhenius rheology as proposed by King [2009]. Here we provided an overview over various nondimensional forms of the Arrhenius formulations, by which we pointed out that quantities as internal temperature or plate velocity and thickness can differ considerably among the different Arrhenius formulations, but can also be very similar to the Frank-Kamenetskii rheology for some formulations. In particular, we observe that the conversion method proposed by Solomatov and Moresi [1996], in which the temperature derivative of the Arrhenius and Frank-Kamenetskii viscosities should be equal, leads to a good similarity between Arrhenius and Frank-Kamenetskii. The cutoff-Arrhenius viscosity is comparable to the derivative-Arrhenius viscosity and thus also very similar to the Frank-Kamenetskii. In contrast to some other Arrhenius formulations, in the cutoff-Arrhenius viscosity both the offset temperature and the activation energy are in a realistic range but too high surface viscosities are cutoff. As King [2009] has shown, results are unchanged as long as the viscosity cutoff is larger than 104 times the bottom viscosity. Additionally, a conversion method proposed by King [2009] results in a good agreement between Arrhenius and Frank-Kamenetskii.

[51] The same general flow behavior for the Frank-Kamenetskii approximation and all Arrhenius rheologies is also obtained for an additional stress dependence of the viscosity: the stagnant lid is deformed allowing for plate-like mobilization. This additional deformation reduces the overall viscosity contrast as the stress-dependent viscosity outweighs the temperature dependence [cf., Stein and Hansen, 2008]. Consequently, the differences among all rheologies are slightly reduced. We find a very good agreement in plate thickness and surface topography. The exact values of the transitions between the flow regimes, the velocities and interior temperatures vary more strongly for each formulation unless properly scaled. When normalizing the Rayleigh number by the critical Rayleigh number, we also obtain a very good agreement in regime transitions for Arrhenius and Frank-Kamenetskii.

[52] To conclude, as so far Earth parameters cannot be reached in numerical studies, approximations to the viscosity have to be made. In this study, we have shown that all commonly applied viscosity laws lead to the same qualitative behavior independent of their parameters (activation energy, offset parameter, conversion method). They can therefore be equally applied for studies of general flow behavior, while quantitative results have to be handled with care. Here we, however, observed that the differences in the quantitative results among all rheologies are reduced when considering the correct scaling and if a more realistic viscosity, depending also on the stress, is employed.


[53] We thank the editor Thorsten Becker as well as Masanori Kameyama, Scott King, Paul Tackley, and an anonymous reviewer for their helpful comments.