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[1] The formation of longest-wavelength mantle convection in the sluggish-lid regime is investigated using a three-dimensional spherical model. The bottom Rayleigh number is fixed at 10^{7}. Considering temperature-dependent rheology, degree-one dominant thermal convection occurs for both purely basal heating and mixed (i.e., basal and internal) heating modes. For the purely basal heating mode, degree-one convection occurs when the viscosity contrast due to temperature-dependent rheology is 10^{3}–10^{4} in both Boussinesq and extended-Boussinesq fluids. However, with extended-Boussinesq fluid, degree-one convection may only occur in the basal heating mode: In the mixed heating mode, degree-one convection shifts to one with high-degree modes, presumably because of enhanced viscous dissipation in the highly viscous lid over up/downwelling plumes. The geophysically relevant degree-two convection with sheet-like downwellings is not observed in this study. The inclusion of visco-plastic rheology in the top part of the mantle breaks down degree-one convection.

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[2] The present-day Earth's mantle convection pattern is certainly dominated by spherical harmonic degrees of two, as illustrated by almost all seismic tomography models [e.g., Ritsema et al., 1999; Masters et al., 2000]. However, it is still a controversial issue whether mantle convection has been dominated by thermal heterogeneity with degree-two or a longer component (i.e., degree-one) throughout the Earth's history.

[3] Previous numerical calculations have shown that degree-one convection is never present in three-dimensional (3-D) spherical models of iso-viscous mantle convection [Schubert et al., 2001]. In recent years, however, degree-one convection has been produced by some numerical models that considered temperature-dependent rheologies in Boussinesq fluid convection. When internal heating is included with moderately temperature-dependent viscosity, degree-one convection occurs [McNamara and Zhong, 2005]. However, Yoshida and Kageyama [2006] have shown that degree-one convection occurs even without internal heating when temperature-dependence of the viscosity is moderately strong. This degree-one convection pattern belongs to the “sluggish-lid” (SL) regime (or transitional regime) [Solomatov, 1993]. Meanwhile [Roberts and Zhong, 2006] have shown that degree-one convection occurs for the “stagnant-lid” regime in extended-Boussinesq (EB) fluid with mixed heating and strongly temperature-dependent viscosity.

[4] Such an inherent convection pattern with the longest-wavelength thermal component may not only affect the reorganization of flow due to supercontinent formation [e.g., Gurnis, 1988] or plate motion [e.g., Bunge and Richards, 1996], but also long-term motion of the true polar wander, which may be associated with mantle convection current [e.g., Greff-Lefftz, 2004] and/or continental drift [Nakada, 2007]. Thus, it is still necessary to investigate the formation of the longest-wavelength convection relevant to the Earth's (or other terrestrial) mantle to provide further insight into mantle dynamics.

[5] In this study, to clarify further whether degree-one and -two convection (i.e., “low-degree convection”) originate in mantle convection in the sluggish-lid regime for both basal heating and mixed (i.e., basal and internal) heating modes and compressibility (including viscous dissipation), numerical calculations have been performed using a 3-D spherical model.

2. Model

[6] The mantle in a 3-D spherical shell of 2867 km thickness is modeled as an infinite Prandtl number fluid using the parallel finite-volume code ConvGS [e.g., Yoshida, 2008], which has been benchmarked extensively. Under the EB fluid approximation [e.g., Christensen and Yuen, 1985], the non-dimensional conservation equation of energy is,

where T is the temperature, t the time, v the velocity vector, v_{r} the radial velocity, and Φ is the viscous dissipation rate per unit volume. The constant ζ ≡ _{1}/, where r_{1} is the Earth's radius, and b is the thickness of the mantle. The Rayleigh number, the dissipation number and the internal heating number are given by Ra ≡ ^{3}/, Di ≡ /_{p}, and Q ≡ ^{2}/_{p}, respectively (Table 1). Here dimensional quantities are indicated with overbars. The non-dimensional viscosity is given in the Arrhenius form,

where E is a free parameter that controls the degree of viscosity's temperature dependence, and T_{top} and T_{bot} are the top and bottom temperatures. This equation implies that the viscosity contrast across the mantle is Δη ≡ exp(E).

The reference viscosity is defined at the bottom boundary. The internal heating rate in “chondritic meteorites” [Turcotte and Schubert, 2002] is referred for extended-Boussinesq cases.

ρ

Density

3300 kgm^{−3}

g

Gravitational acceleration

9.8 ms^{−2}

α

Thermal expansivity

1.6 × 10^{−5} K^{−1}

ΔT

Temperature difference

2500 K

T_{top}

Top temperature

250 K

b

Mantle thickness

2867 km

κ

Thermal diffusivity

10^{−6} m^{2}s^{−1}

η

Viscosity

3 × 10^{21} Pa s

c_{p}

Specific heat

1250 Jkg^{−1}K^{−1}

H

Internal heating rate

3.5 × 10^{−12} Wkg^{−1}

[7] Impermeable, shear-stress-free, and iso-thermal conditions are adopted on both top and bottom boundaries. For all calculations, the number of grid points used was 102(r) × 102(θ) × 302 (ϕ) × 2 (two component grids). All the models have been calculated for at least 0.3 thermal diffusion time before a statistically steady state is reached.

3. Results

[8] The first iso-viscous, Boussinesq (i.e., Di = 0) and EB (Di = 0.36) models are initiated with random temperature perturbations. For each model with the temperature-dependent rheology, E is systematically changed so as to increase Δη by 10^{0.5} from Δη = 10^{0} to Δη = 10^{6}. In this study, Ra is fixed at 1.0 × 10^{7} using the physical values in Table 1. The dependence of Ra on the convection pattern is discussed in Section 4.

[9] We start with a Boussinesq fluid model (Figure 1). When the layer is bottom-heated mantle (Q = 0) and Δη = 10^{2} (E ∼ 4.61), the convection pattern is in the SL regime and dominated by a spherical harmonic degree (l) of 2 with two cylindrical downwellings surrounded by a sheet-like upwelling (Figure 1a), corroborating a result of Ratcliff et al. [1997]. When the viscosity contrast is Δη = 10^{4} (E ∼ 9.21), the convection pattern is still in the SL regime, but becomes l = 1 dominant (Figures 1b and 1e). When Δη ≥ 10^{5}, convection falls into the stagnant-lid regime in which an immobile highly viscous layer is formed at the top boundary as shown in the previous models with basal [Ratcliff et al., 1997; Yoshida and Kageyama, 2006] and mixed [Roberts and Zhong, 2006] heating mode.

[10] Next, in addition to the bottom heating, internal heating (Q = 10) is considered. The convection shifts to one with high-degree modes when Δη = 10^{2} (Figure 1c) to the degree-one pattern when Δη = 10^{4} (Figure 1d). Consequently, when Δη = 10^{3} − 10^{4}, degree-one convection appears in both bottom heating and mixed heating models. Cross-sections show that the surface is covered by a slowly moving lid (sluggish-lid) and the downwelling plumes develop from the lid. As a result, globally large-scale flow develops in the mantle interior (Figures 1e and 1f).

[11] To investigate the effects of compressibility (including viscous dissipation) on low-degree convection, EB calculations with Di = 0.36 were performed (Figure 2). In the purely bottom heated mantle (Q = 0), the degree-two pattern changes into a new pattern dominated by l = {2, 3, 5}, characterized by three columnar downwellings and a network of sheet-like upwellings (Figure 2a). However, when Δη = 10^{4} (Figure 2b), thermal structures of convection do not change from the Boussinesq case (Figure 1b) and retain the degree-one pattern. On the other hand, when internal heating is considered (Q = 9.2), the convection patterns change from Boussinesq cases. When Δη = 10^{4}, the pattern shifts to one with high-degree modes because the bottom of the highly viscous lid becomes unsteady presumably with enhanced viscous dissipation heating (Figure 2d); viscous heating is relatively large at the top of up/downwelling plumes and where the plumes impinge on the highly viscous lid. This trend for high-degree modes increases when Δη = 10^{2}; the lower-degree modes (l ≤ 12) have now almost vanished (Figure 2c).

[12] It is known that viscous dissipation heating is relatively large near the upper and lower boundaries comparable to the interior of the convection cell [Zhang and Yuen, 1996]. While the influence of both compressibility and viscous dissipation on the form of global mantle convection is relatively minor without temperature-dependent rheology [Schubert et al., 2001], it has been shown that the long-wavelength thermal structure, particularly at degree-one is significantly modified by compressibility and viscous dissipation.

[13] It is an open question whether different surface conditions modify the degree-one convection. I now impose a visco-plastic rheology [e.g., Moresi and Solomatov, 1998] at the top part (above 250 km depth) of the model with bottom heating, Boussinesq approximation, and Δη = 10^{3}. Figures 3a and 3b illustrate the viscosity and velocity fields at the surface for models with the yield stress (τ_{y}) of 100 MPa and 50 MPa (the corresponding “differential stresses” are 200 MPa and 100 MPa, respectively). When τ_{y} = 100 MPa, the low-viscosity convergence zone (see a red arrow) is not perfectly covered across the entire surface and the mantle structure still retains the degree-one pattern (Figure 3a). On the other hand, the relatively lower level of the yield stress (τ_{y} = 50 MPa) results in the development of long, linear convergence zones in the highly viscous lid (Figure 3b) and short-wavelength heterogeneity with sheet-like downwellings in the mantle (Figure 3c).

4. Discussion and Conclusions

[14] This study focuses on the formation of low-degree convection in the sluggish-lid (SL) regime with 3-D spherical models. For the bottom heating mode, degree-one dominant convection is found when the viscosity contrast due to temperature-dependent rheology is 10^{3}–10^{4} for both Boussinesq and EB fluids. For the mixed heating mode, a degree-two dominant convection was not observed in the parameter space searched in this study, corroborating previous work [McNamara and Zhong, 2005, Table 1]. However, it is noted that the convection pattern is likely to be very sensitive to Ra, as implied by the work of McNamara and Zhong [2005]. For instance, it was confirmed that when Ra = 4.56 × 10^{5}, Q = 0 and η = exp[(ln 10^{4}) (0.5 − T)] are taken in Boussinesq fluid, as studied by McNamara and Zhong [2005] (that is, effective Ra at the bottom is 4.56 × 10^{7}), the convection pattern becomes the short-wavelength thermal structure, in contrast to degree-one convection at Ra = 10^{7} (Figure 1b). Furthermore, in fact, although Yoshida and Kageyama [2006] have found that the Earth-like degree-two pattern characterized by two columnar upwellings and one sheet-like downwelling appear when the Rayleigh number is relatively low (Ra = 10^{6}), such degree-two convection never occurred with more realistic Rayleigh number (Ra = 10^{7}) in this study.

[15]Roberts and Zhong [2006] have shown that degree-one convection occurs for the stagnant-lid regime in EB fluid with mixed heating. Also Zhong et al. [2007] have found that for a mantle with a highly viscous lower mantle and lithosphere, a weak upper mantle or asthenosphere is needed to realize degree-one convection. In the current study, it has been shown that degree-one convection may not occur in EB fluid in the sluggish-lid regime, but the current study did not include radial viscosity variations.

[16] It is suggested that over the Earth's history, if internal heating becomes too small and the viscosity contrast between mantle boundaries becomes too large, the convection may reach a degree-one pattern as a final state. It has also been confirmed that for the SL regime, both degree-one and degree-two dominant convection are obtained only in the purely bottom heated mantle, depending upon Δη. Although it is uncertain how much the viscosity contrast increased or decreased throughout Earth's history, it seems that the large fraction of basal heating favors the change of its convection pattern in the SL regime. Convection with the strong bottom heat flow is supported by recent 2-D mantle modelings [e.g., Yoshida and Ogawa, 2005; Mittelstadt and Tackley, 2006], seismological observation [Lay et al., 2006; van der Hilst et al., 2007], and geodynamo energetics [Lay et al., 2008]. This is in contrast to an earlier inference that suggested a weak CMB heat flow [e.g., Davies, 1999]. More recently, using a 3-D spherical model with adiabatic heating, CMB heat flux inferred from Zhong [2006] and Leng and Zhong [2008] accounts for ∼35–40% of surface heat flux.

[17] Once it forms, low-degree convection is in a nearly steady state in time, as the flow pattern is mostly determined by a small deformation of the top thermal/rheological boundary layer. Such a robust thermal structure may be modified by imposing different surface conditions, such as supercontinents and/or plate motion. Zhong et al. [2007] have shown the cyclic process between degree-one and degree-two convection by incorporating a highly viscous super-continent. In this study, the potential effect of surface plate-like motion on the breakdown of degree-one convection has been explored. The results indicate that the subduction zones surrounding the margins of the past supercontinents since the Palaeozoic era (i.e., Rodinia and Pangaea), as seen in some paleographic reconstructions [e.g., Collins, 2003; Torsvik, 2003], may be an inherent characteristic of mantle convection only with thermally induced driving force (that is, without any chemical heterogeneity like the compositionally distinct continent).

Acknowledgments

[18] The author is grateful to Michael Gurnis and an anonymous reviewer for their careful reviews and thoughtful comments which greatly benefited the paper. Calculations were performed on the supercomputer facilities (SGI Altix 4700) of JAMSTEC. This study was supported by a Grant-in-Aid for Scientific Research on Priority Areas (16075205) and for Young Scientists (B) (20740260) by MEXT of Japan.