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[1] Despite the tight constraints put by seismology on the elastic properties of the Earth's lower mantle, its mineralogical composition and thermal state remain poorly known because the interpretation of seismic measurements suffers from the trade-off between temperature, iron content, and mineralogical composition. In order to overcome this difficulty, we complement seismic data with electromagnetic induction data. The latter data are mostly sensitive to temperature and iron content, while densities and acoustic speeds mostly constrain the mineralogy. A 0.5 log unit increase in electrical conductivity can be caused either by a 400 K increase of the temperature or by an increase of iron content from 10% to 12.5%. Acoustic velocity is only marginally sensitive to temperature but it increases by 0.8 km s À1 on average as the perovskite fraction increases from 50% to 100%. Olsen's (1999) apparent resistivities in the period range [15 days, 11 years], and Preliminary reference Earth model (PREM) densities and acoustic speeds are jointly inverted in the depth range [800 km, 2600 km] by using a Monte Carlo Markov Chain method. Given the uncertainties on these data, estimates of perovskite fraction are well constrained over the whole depth range, but information on temperature and iron content is only obtained for depths less than 2000 km, corresponding to the penetration depth of the long-period electromagnetic field. All parameter values are determined with an uncertainty better than 15-20% at the 1s confidence level. The temperature in the uppermost lower mantle (i.e

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how the discretization of the governing equations affects the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in developing mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes -an established, communitydeveloped, and benchmarked finite-element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBFs) -are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree = 4). How this pattern varies to perturbations in the initial condition and Rayleigh number is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree = 6). Although both methods first converge to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady-state pattern is presented as a combination of third-and fourth-order spherical harmonics leading to a fivecell or hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.

Abstract. Fully 3-D numerical simulations of thermal convection in a spherical shell have become a standard for studying the dynamics of pattern formation and its stability under perturbations to various parameter values. The question arises as to how does the discretization of the governing equations affect the outcome and thus any physical interpretation. This work demonstrates the impact of numerical discretization on the observed patterns, the value at which symmetry is broken, and how stability and stationary behavior is dependent upon it. Motivated by numerical simulations of convection in the Earth's mantle, we consider isoviscous Rayleigh-Bénard convection at infinite Prandtl number, where the aspect ratio between the inner and outer shell is 0.55. We show that the subtleties involved in development mantle convection models are considerably more delicate than has been previously appreciated, due to the rich dynamical behavior of the system. Two codes with different numerical discretization schemes: an established, community-developed, and benchmarked finite element code (CitcomS) and a novel spectral method that combines Chebyshev polynomials with radial basis functions (RBF) are compared. A full numerical study is investigated for the following three cases. The first case is based on the cubic (or octahedral) initial condition (spherical harmonics of degree ℓ =4). How variations in the behavior of the cubic pattern to perturbations in the initial condition and Rayleigh number between the two numerical discrezations is studied. The second case investigates the stability of the dodecahedral (or icosahedral) initial condition (spherical harmonics of degree ℓ = 6). Although both methods converge first to the same pattern, this structure is ultimately unstable and systematically degenerates to cubic or tetrahedral symmetries, depending on the code used. Lastly, a new steady state pattern is presented as a combination of order 3 and 4 spherical harmonics leading to a five cell or a hexahedral pattern and stable up to 70 times the critical Rayleigh number. This pattern can provide the basis for a new accuracy benchmark for 3-D spherical mantle convection codes.