Exploring the physics of the Earth’s core with numerical simulations Nathanaël Schaeffer To cite this version: Nathanaël Schaeffer. Exploring the physics of the Earth’s core with numerical simulations. Geophysics [physics.geo-ph]. Grenoble 1 UJF - Université Joseph Fourier; Université Grenoble Alpes, 2015. tel- 01241755 HAL Id: tel-01241755 https://tel.archives-ouvertes.fr/tel-01241755 Submitted on 11 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial - ShareAlike| 4.0 International License Habilitation a` Diriger des Recherches Universite´ Grenoble Alpes Exploring the physics of the Earth’s core with numerical simulations Nathanael¨ Schaeffer soutenue le 30 septembre 2015, devant le jury compos´ede : Emmanuel DORMY Ecole´ Normale Sup´erieure(Paris) Rapporteur B´ereng`ereDUBRULLE Centre d’Etudes´ Atomiques (Saclay) Rapporteur Johannes WICHT Max Planck Institute (G¨ottingen) Rapporteur Thierry ALBOUSSIERE` Universit´eClaude Bernard (Lyon) Examinateur Franck PLUNIAN Universit´eGrenoble Alpes (Grenoble) Examinateur Abstract In the first chapter of this report, I discuss some of my work of the past 7 years, since I joined the geodynamo team at ISTerre as a CNRS researcher. This work most often involves numerical simulations with codes that I have written. An important step forward in the efficiency of simulations based on the spher- ical harmonic transform has come from the matrix-free SHTns library I have de- signed and written (Schaeffer, 2013). Numerical simulations linked to the magnetized spherical Couette experiment DTS have been performed to understand its peculiar turbulence (Figueroa et al., 2013) and try to characterize the effect of the small turbulent scales on the in- duction processes (Cabanes, Schaeffer, and Nataf, 2014a; Cabanes, Schaeffer, and Nataf, 2014b). Related to the Earth’s core dynamics, my simulations helped to characterize the effects of the magnetic field on short timescale flows, leading to strong ar- guments for quasi-geostrophic (columnar) flows at large spatial scales (& 10 km) and short timescales (. 10 years) in the core (Gillet, Schaeffer, and Jault, 2011). Smaller length scales evade the rotational constraint because inertial waves are getting too slow. At longer length scales more research is needed, but we have already explored the implications of a deformation of the columns by magnetic fields (Schaeffer, Lora Silva, and Pais, 2016). We have also shown that near the equator, where the columnar flows were expected to wither, the quasi-geostrophy seems strong in the Earth’s core (Schaeffer and Pais, 2011). Prompted by observations, we studied the propagation and reflection of tor- sional Alfv´enwaves in the core, and showed the importance of the value of the magnetic Prandtl number (Schaeffer, Jault, et al., 2012). Moreover, an extension of this work to include a conducting solid layer at the top of the core suggests the existence of such a layer with a rather strong conductance in the Earth. More recently, and as a logical follow-up, I have produced turbulent geody- namo simulations, with interesting implications that will be reported in a future publication, but several facts are already presented in appendixD and will also be discussed here. These simulations have also fed the reflection that resulted in our chapter on core turbulence in the second edition of the Treatise on Geophysics (Nataf and Schaeffer, 2015). In the second chapter, I present synthetically my ongoing work and projects that develop even further the topics above, but also new projects on the dynamo of the early Moon and further numerical developments. Contents 1 Research summary of selected work4 1.1 Introduction............................4 1.1.1 Motivation.........................4 1.1.2 Governing equations...................5 1.1.3 Control Parameters....................6 1.1.4 Diagnostic Parameters..................7 1.1.5 Open questions...................... 10 1.2 Numerical simulations in spherical geometry.......... 11 1.2.1 Is it possible to improve the SHT?........... 12 1.2.2 Improving the performance of spherical codes..... 13 1.3 Simulations of the DTS experiment............... 15 1.4 Simulations motivated by observations............. 18 1.4.1 Reflection of Alfv´enwaves................ 18 1.4.2 Quasi-geostrophic flows in the Earth’s core?...... 24 1.5 Turbulence in the Core...................... 29 1.5.1 Waves........................... 29 1.5.2 Inferred regimes for the core............... 31 1.5.3 Turbulence in geodynamo simulations?......... 31 2 Research projects and ongoing work 36 2.1 Understanding Turbulent dynamos............... 36 2.1.1 Force balances....................... 36 2.1.2 Mean fields........................ 38 2.1.3 Can we do better?.................... 38 2.2 MagLune: the dynamo of the Moon............... 39 2.2.1 Convective dynamos................... 39 2.2.2 Precession dynamos in spherical shells......... 40 2.3 Simulation of experiments.................... 41 2.3.1 DTS............................ 41 2.3.2 ZoRo............................ 42 2.4 Eigenmodes with magnetic fields................. 42 1 2.5 From spheres to ellipsoids.................... 43 2.6 Improving the numerical methods................ 44 2.6.1 More flexible implicit time-schemes........... 44 2.6.2 Matrix-free implicit methods............... 45 A Vitae 46 A.1 Teaching & advising....................... 47 A.1.1 Other responsibilities................... 48 A.2 Publications............................ 48 A.2.1 Refereed articles..................... 48 A.2.2 Book chapter....................... 50 A.2.3 Invited talks........................ 50 A.2.4 Software.......................... 50 A.2.5 Participation to conferences............... 51 B Selection of papers 52 B.1 Rationale and geophysical evidence for quasi-geostrophic rapid dynamics within the Earth’s outer core............. 53 B.2 On symmetry and anisotropy of Earth-core flows........ 64 B.3 On the reflection of Alfv´enwaves and its implication for Earth’s core modelling........................... 69 B.4 Efficient Spherical Harmonic Transforms aimed at pseudo- spectral numerical simulations.................. 78 B.5 Modes and instabilities in magnetized spherical Couette flow. 89 B.6 Turbulence Reduces Magnetic Diffusivity in a Liquid Sodium Experiment............................ 114 B.7 Turbulence in the Core...................... 120 B.8 Can Core Flows inferred from Geomagnetic Field Models ex- plain the Earth’s Dynamo?.................... 142 C The XSHELLS code 162 C.1 Description............................ 162 C.2 Framework............................. 163 C.3 Boundary conditions....................... 163 C.3.1 Central condition at r = 0................ 163 C.3.2 Magnetic field....................... 164 C.3.3 Temperature field..................... 166 C.3.4 Velocity field....................... 168 C.4 Time-stepping scheme...................... 170 C.4.1 Adams-Bashforth with variable time-step........ 171 C.5 Implementation of variable conductivity............ 171 2 C.5.1 Induction equation.................... 171 C.5.2 Continous variation with radius r ............ 171 C.5.3 Radial discontinuities of the conductivity........ 173 C.6 Optimization and parallelization................. 175 C.6.1 hybrid MPI/OpenMP parallelization.......... 175 C.6.2 blocked LU-solver..................... 176 D A gallery of turbulent geodynamo simulations 178 D.1 Overview.............................. 178 D.2 Instant fields........................... 181 D.3 Time averaged fields....................... 185 Bibliography 187 3 Chapter 1 Research summary of selected work 1.1 Introduction 1.1.1 Motivation Describing and understanding the Deep Earth The Earth deep interior is impossible to observe directly, but it is possible to probe it using several tools. The seismic waves traveling through our planet yield most of the information we have on its structure, through the travel speed and reflection at the interfaces of the compression and shear waves, but also through the frequency and attenuation of the seismic normal modes. The Earth’s rotation variations, which respond to gravitational torques, depend not only on its moment of inertia (the mass distribution), but also whether there are fluid layers and the coupling between the fluid layers and the solid ones. A good example of the usefulness of these measurements is the success of the PREM model (Dziewonski and Anderson, 1981), which included data from seismology (both travel time and normal modes) as well as total mass and moment of inertia. The magnetic field of our planet is produced by fluid motion in its core, and as such it gives away information about the flow in the core. This flow is arguably driven by convection due to the slow cooling of the Earth. It is also influenced by the magnetic field itself, but also by some details of its boundaries. It is also important to keep in mind that if the mantle near the core-mantle boundary is slightly conducting, there will be some electro- magnetic coupling of the core