# The Bunge et al. mantle convection experiments#

This section was contributed by Cedric Thieulot and Bob Myhill.

Early mantle modeling studies of the 1970s, 1980s and 1990s were often concerned with simple set-ups (Cartesian geometries, incompressible fluid, free slip boundary conditions) and investigated the influence of the Rayleigh number, the heating mode or the temperature dependence of the viscosity on temperature, pressure and/or strain rate (Young 1974; F. H. Busse 1975; FH Busse 1979; Blankenbach et al. 1989; F. Busse et al. 1993; Berg, Keken, and Yuen 1993; Bunge, Richards, and Baumgardner 1997). In this cookbook, we use the ‘simple’ material model to reproduce the set-up in (Bunge, Richards, and Baumgardner 1996), which reported that even modest increases in mantle viscosity with depth could have a marked effect on the style of mantle convection. The prm file corresponding to this cookbook can be found at cookbooks/bunge_et_al_mantle_convection/bunge_et_al.prm.

Although the original article showcases results obtained in a 3D hollow sphere, we here run the models in an annular domain of inner radius $$R\_{inner} = 3480~\si{\km}$$ and outer radius $$R\_{outer} = 6370~\si{\km}$$. The surface temperature is set to $$T{\_{surf}}$$ = 1060 K and the bottom temperature to $$T{\_{cmb}} = 3450$$ K. The gravity vector is radial and its magnitude is $$g = 10$$ m s−2.

There is a single incompressible fluid in the domain, characterized by $$\rho_0 = 4500$$ kg m−3, $$\alpha = 2.5\cdot10^{-5}$$  K, $$k = 4$$ W m K−−1, $$C_p = 1000$$ J kg K−−1 and its internal heating rate is $$Q{\_{int}} = 1\cdot10^{-12}$$ W kg−1. The interface between the upper mantle (viscosity $$\eta\_{um}$$) and the lower mantle (viscosity $$\eta\_{lm}$$) is fixed at 670 km depth. As in the article we consider four time-independent radial viscosity profiles:

• Isoviscous mantle: $$\eta\_{um}=\eta\_{lm}=1.7\cdot 10^{24}$$ Pa s

• Mantle with step change in viscosity: $$\eta\_{um}=5.8\cdot 10^{22}$$ Pa s, $$\eta\_{lm}=30\eta\_{um}$$

• Isoviscous mantle: $$\eta\_{um}=\eta\_{lm}=5.8\cdot 10^{22}$$ Pa s

• Mantle with step change in viscosity: $$\eta\_{um}=7\cdot 10^{21}$$ Pa s, $$\eta\_{lm}=30\eta\_{um}$$

Separate ascii files visc_depth_X.txt with X={a,b,c,d} contain each of these viscosity profiles. The resulting temperature fields after 5 billion years of convection are shown in Fig. 1. Similar to the results obtained by (Bunge, Richards, and Baumgardner 1996), models in which the lower mantle is more viscous than the upper mantle are distinctly colder than their isoviscous equivalents, with more clearly defined upwellings. You can find a movie of how the temperature evolves over this time period at https://youtu.be/5SPCU1sFGGc.

Berg, Arie P van den, Peter E van Keken, and David A Yuen. 1993. “The Effects of a Composite Non-Newtonian and Newtonian Rheology on Mantle Convection.” Geophy. J. Int. 115 (1): 62–78.

Blankenbach, B., F. Busse, U. Christensen, L. Cserepes, D. Gunkel, U. Hansen, H. Harder, et al. 1989. “A Benchmark Comparison for Mantle Convection Codes.” Geophys. J. Int. 98: 23–38.

Bunge, H.-P., M. A. Richards, and J. R. Baumgardner. 1996. “Effect of depth-dependent viscosity on the planform of mantle convection.” Nature 379: 436–38.

———. 1997. “A sensitivity study of three-dimensional spherical mantle convection at $$10^8$$ Rayleigh number: Effects of depth-dependent viscosity, heating mode, and endothermic phase change.” J. Geophys. Res. 102 (B6): 11, 991–12, 007. https://doi.org/10.1029/96JB03806.

Busse, F. H. 1975. “Patterns of convection in spherical shells.” J. Fluid Mech. 72 (1): 67–85.

Busse, F., U. Christensen, R. Clever, L. Cserepes, C. Gable, E. Giannandrea, L. Guillou, et al. 1993. “3d Convection at Infinite Prandtl Numbers in Cartesian Geometry — a Benchmark Comparison.” Geophys. Astrophys. Fluid Dynamics 75: 39–59.

Busse, FH. 1979. “High Prandtl Number Convection.” Phys. Earth. Planet. Inter. 19 (2): 149–57.

Young, Richard E. 1974. “Finite-Amplitude Thermal Convection in a Spherical Shell.” J. Fluid Mech. 63 (4): 695–721.