We present an attempt to reach realistic turbulent regime in direct numerical simulations of the geodynamo. We rely on a sequence of three convection-driven simulations in a rapidly rotating spherical shell. The most extreme case reaches towards the Earth's core regime by lowering viscosity (magnetic Prandtl number P m = 0.1) while maintaining vigorous convection (magnetic Reynolds number Rm > 500) and rapid rotation (Ekman number E = 10 −7 ), at the limit of what is feasible on today's supercomputers. A detailed and comprehensive analysis highlights several key features matching geomagnetic observations or dynamo theory predictions -all present together in the same simulation -but it also unveils interesting insights relevant for Earth's core dynamics.In this strong-field, dipole-dominated dynamo simulation, the magnetic energy is one order of magnitude larger than the kinetic energy. The spatial distribution of magnetic intensity is highly heterogeneous, and a stark dynamical contrast exists between the interior and the exterior of the tangent cylinder (the cylinder parallel to the axis of rotation that circumscribes the inner core).In the interior, the magnetic field is strongest, and is associated with a vigorous twisted polar vortex, whose dynamics may occasionally lead to the formation of a reverse polar flux patch at the surface of the shell. Furthermore, the strong magnetic field also allows accumulation of light material within the tangent cylinder, leading to stable stratification there. Torsional Alfvén waves are frequently triggered in the vicinity of the tangent cylinder and propagate towards the equator.Outside the tangent cylinder, the magnetic field inhibits the growth of zonal winds and the kinetic energy is mostly non-zonal. Spatio-temporal analysis indicates that the low-frequency, non-zonal flow is quite geostrophic (columnar) and predominantly large-scale: an m=1 eddy spontaneously emerges in our most extreme simulations, without any heterogeneous boundary forcing.Our spatio-temporal analysis further reveals that (i) the low-frequency, largescale flow is governed by a balance between Coriolis and buoyancy forces -magnetic field and flow tend to align, minimizing the Lorentz force; (ii) the high-frequency flow obeys a balance between magnetic and Coriolis forces; (iii) the convective plumes mostly live at an intermediate scale, whose dynamics is driven by a 3-term 1 arXiv:1701.01299v3 [physics.geo-ph] 15 Jun 2017 MAC balance -involving Coriolis, Lorentz and buoyancy forces. However, smallscale ( E 1/3 ) quasi-geostrophic convection is still observed in the regions of low magnetic intensity.

[1] In this paper, we report on very efficient algorithms for spherical harmonic transform (SHT). Explicitly vectorized variations of the algorithm based on the Gauss-Legendre quadrature are discussed and implemented in the SHTns library, which includes scalar and vector transforms. The main breakthrough is to achieve very efficient on-the-fly computations of the Legendre-associated functions, even for very high resolutions, by taking advantage of the specific properties of the SHT and the advanced capabilities of current and future computers. This allows us to simultaneously and significantly reduce memory usage and computation time of the SHT. We measure the performance and accuracy of our algorithms. Although the complexity of the algorithms implemented in SHTns are in O N 3 ð Þ(where N is the maximum harmonic degree of the transform), they perform much better than any third-party implementation, including lower-complexity algorithms, even for truncations as high as N = 1023. SHTns is available at https://bitbucket.org/nschaeff/shtns as open source software.