Low-degree mantle convection with strongly temperature- and   depth-dependent viscosity in a three-dimensional spherical shell
  A series of numerical simulations of thermal convection of Boussinesq fluid with infinite Prandtl number, with Rayleigh number $10^7$, and with the strongly temperature- and depth- dependent viscosity in a three-dimensional spherical shell is carried out to study the mantle convection of single-plate terrestrial planets like Venus or Mars without an Earth-like plate tectonics. The strongly temperature-dependent viscosity (the viscosity contrast across the shell is $\geq 10^5$) make the convection under stagnant-lid short-wavelength structures. Numerous, cylindrical upwelling plumes are developed because of the secondary downwelling plumes arising from the bottom of lid. This convection pattern is inconsistent with that inferred from the geodesic observation of the Venus or Mars. Additional effect of the stratified viscosity at the upper/lower mantle (the viscosity contrast is varied from 30 to 300) are investigated. It is found that the combination of the strongly temperature- and depth-dependent viscosity causes long-wavelength structures of convection in which the spherical harmonic degree $\ell$ is dominant at 1--4. The geoid anomaly calculated by the simulated convections shows a long-wavelength structure, which is compared with observations. The degree-one ($\ell = 1$) convection like the Martian mantle is realized in the wide range of viscosity contrast from 30 to 100 when the viscosity is continuously increased with depth at the lower mantle. 
Masaki Yoshida, Akira Kageyama