**A new method to simulate convection with strongly temperature- and pressure-dependent voscosity in a spherical shell: Applications to the Earth's mantle**

*Kai Stemmer, Helmut Harder, Ulrich Hansen *

We present a new finite volume code for
modeling three-dimensional thermal convection in a spherical shell with
strong temperature- and pressure-dependent viscosity. A new
discretization formulation of the viscous term, tailored to the finite
volume method on a colocated grid, enables laterally variable
viscosity. A smoothed cubed–sphere grid is used to avoid pole problems
which occur in latitude–longitude grids with spherical coordinates. The
spherical shell is topologically divided into six cubes. The equations
are formulated in primitive variables, and are treated in the Cartesian
cubes. In order to ensure mass conservation a SIMPLER pressure
correction procedure is applied and to handle strong viscosity
variations of Δη = 10^{7} and high Rayleigh numbers of Ra = 10^{8}
the pressure correction algorithm is combined with a pressure weighted
interpolation method to satisfy the incompressibility condition and to
avoid oscillatory pressure solutions. The model is validated by a
comparison of diagnostical parameters of steady-state cubic and
tetrahedral convection with other published spherical models and a
detailed convergence test on successively refined grids. Lateral
variable fluid properties have a significant influence on the
convection pattern and heat flow dynamics. The influence of
temperature- and pressure-dependent viscosity on the flow is
systematically analyzed for basal and mixed-mode heated thermal
convection in the spherical shell. A new method to classify the
simulations to the mobile, transitional or stagnant-lid regime is given
by means of a comparison of selected diagnostical parameters, a
significantly improved classification as compared to the common surface
layer mobility criterion. A scaling law for the interior temperature
and viscosity in the stagnant-lid regime is given. Purely basal heating
and strongly temperature-dependent rheology stabilize plume positions
and yield with a weak time dependence of the convecting system, while
the amount of additional internal heating controls the strength of time
dependence. Strength and partitioning of basal and internal heat
sources in the mantle seems to be of major importance to specify the
dynamics of the flow field and therefore the evolution of the Earth and
other planets. Additional pressure dependence strongly influences the
dynamics even if the magnitude of pressure variation is relatively
small. For an appropriate combination of pressure and temperature
dependence we observe a kind of low and high viscosity zone in the
asthenosphere and deep in the mantle. The viscosity–depth profile of
such a flow shows striking similarities to viscosity profiles from
inversion of seismic, geoid and post-glacial rebound data.