A linear stability analysis on the onset of thermal convection of a fluid with strongly temperature-dependent viscosity in a spherical shell

  • Masanori Kameyama
  • Hiroki Ichikawa
  • Arata Miyauchi
Open Access
Original Article


A linear stability analysis was performed in order to study the onset of thermal convection in the presence of a strong viscosity variation, with a special emphasis on the condition for the stagnant-lid (ST) convection where a convection takes place only in a sublayer beneath a highly viscous lid of cold fluid. We consider the temporal evolution (growth or decay) of an infinitesimal perturbation superimposed to a Boussinesq fluid with an infinite Prandtl number which is in a static (motionless) and conductive state in a basally heated planar layer or spherical shell. The viscosity of the fluid is assumed to be exponentially dependent on temperature. The linearized equations for conservations of mass, momentum, and internal (thermal) energy are numerically solved for the critical Rayleigh number, Ra c , as well as the radial profiles of eigenfunctions for infinitesimal perturbations. The above calculations are repeatedly carried out by systematically varying (i) the magnitude of the temperature dependence of viscosity, E, and (ii) the ratio of the inner and outer radii of the spherical shell, γ. A careful analysis of the vertical structure of incipient flows demonstrated that the dominance of the ST convection can be quantitatively identified by the vertical profile of Δ h (a measure of conversion between horizontal and vertical flows), regardless of the model geometries. We also found that, in the spherical shell relevant to the Earth’s mantle (γ = 0.55), the transition into ST convection takes place at the viscosity contrast across the layer \({r_\eta\simeq10^4}\) . Taken together with the fact that the threshold value of r η falls in the range of r η for a so-called sluggish-lid convection, our finding suggests that the ST-mode of convection with horizontally elongated convection cells is likely to arise in the Earth’s mantle solely from the temperature-dependent viscosity.


Mantle convection Temperature-dependent viscosity Spherical shell Linear stability analysis Stagnant-lid convection 



We thank Drs. Takashi Nakagawa, Shijie Zhong, and David A. Yuen for fruitful discussion. We also thank Sabrina Whitaker for correcting the grammatical and typological errors in the manuscript. We thank three anonymous reviewers for valuable comments which greatly improved the manuscript. We greatly acknowledge thorough support from the Global COE program from the Ministry of Education, Culture, Sports, and Technology (MEXT) of Japan. MK acknowledges financial supports from Grant-in-Aid for Scientific Research (C; #19540452) and (B; #22340127) from Japan Society for the Promotion of Science.

Open Access

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© The Author(s) 2011

Authors and Affiliations

  • Masanori Kameyama
    • 1
  • Hiroki Ichikawa
    • 1
  • Arata Miyauchi
    • 1
  1. 1.Geodynamics Research CenterEhime UniversityMatsuyamaJapan

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