# Nathanaël SchaefferFrench National Centre for Scientific Research | CNRS · Institut des sciences de la Terre (ISTerre)

Nathanaël Schaeffer

PhD, HDR

## About

64

Publications

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1,458

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## Publications

Publications (64)

In this paper, we report on very efficient algorithms for the 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 compu...

Torsional Alfv{\'e}n waves propagating in the Earth's core have been inferred by inversion techniques applied to geomagnetic models. They appear to propagate across the core but vanish at the equator, exchanging angular momentum between core and mantle. Assuming axial symmetry, we find that an electrically conducting layer at the bottom of the mant...

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 Pm=0.1) while maintaining vigorou...

Convection is a fundamental physical process in the fluid cores of planets. It is the primary transport mechanism for heat and chemical species and the primary energy source for planetary magnetic fields. Key properties of convection—such as the characteristic flow velocity and length scale—are poorly quantified in planetary cores owing to the stro...

Earth’s magnetic field is generated by fluid motions in the outer core. This geodynamo has operated for over 3.4 billion years. However, the mechanism that has sustained the geodynamo for over 75% of Earth’s history remains debated. In this Review, we assess the mechanisms proposed to drive the geodynamo (precession, tides and convection) and their...

Understanding fluid flows in planetary cores and subsurface oceans, as well as their signatures in available observational data (gravity, magnetism, rotation, etc.), is a tremendous interdisciplinary challenge. In particular, it requires understanding the fundamental fluid dynamics involving turbulence and rotation at typical scales well beyond our...

The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using bo...

The generation of mean flows is a long-standing issue in rotating fluids. Motivated by planetary objects, we consider here a rapidly rotating fluid-filled spheroid, which is subject to weak perturbations of either the boundary (e.g. tides) or the rotation vector (e.g. in direction by precession, or in magnitude by longitudinal librations). Using bo...

Convection in a spherical shell is widely used to model fluid layers of planets and stars. The choice of thermal boundary conditions in such models is not always straightforward. To understand the implications of this choice, we report on the effects of the thermal boundary condition on thermal convection, in terms of instability onset, fully devel...

In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection (DDC). We investigate rotating DDC (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal thermal and compo...

Precession of planets or moons affects internal liquid layers by driving flows, instabilities and possibly dynamos. The energy dissipated by these phenomena can influence orbital parameters such as the planet’s spin rate. However, there is no systematic study of these flows in the spherical shell geometry relevant for planets, and the lack of scali...

Big Challenges on Joliot-Curie - Before it goes in full production, a series of Grand Challenges – large scale simulations from academic and industry research – were carried out on Joliot-Curie supercomputer, from June to September 2018, to check that the system is functioning properly. These Grand Challenges represent a unique opportunity for sele...

The discovery of torsional Alfvén waves (geostrophic Alfvén waves) in the Earth’s core (Gillet et al. 2010) calls for a better understanding of their properties. We present the first experimental observations of torsional Alfvén waves, performed in the DTS-Ω set-up. In this set-up, 50 L of liquid sodium are confined between an inner sphere (ri = 74...

In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection. We investigate rotating double-diffusive convection (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal...

High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approx...

The current trend in high-performance computing of a stagnating or even decreasing processor speed poses new challenges to solve PDEs within a particular time frame. Here, disruptive mathematical reformulations which exploit additional degrees of parallelism also in the time dimension gained increasing interest over the last two decades.
One of su...

Convection is a fundamental physical process in the fluid cores of planets because it is the primary transport mechanism for heat and chemical species and the primary energy source for planetary magnetic fields. Key properties of convection, such as the characteristic flow velocity and lengthscale, are poorly quantified in planetary cores due to th...

Precession of planets or moons affects internal liquid layers by driving flows, instabilities and possibly dynamos.The energy dissipated by these phenomena can influence orbital parameters such as the planet's spin rate.However, there is no systematic study of these flows in the spherical shell geometry relevant for planets, turning any extrapolati...

High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approx...

The Derviche Tourneur sodium experiment, a spherical Couette magnetohydrodynamics experiment with liquid sodium as the medium and a dipole magnetic field imposed from the inner sphere, recently underwent upgrades to its diagnostics to better characterize the flow and induced magnetic fields with global rotation. In tandem with the upgrades, a set o...

Stellar magnetism plays an important role in stellar evolution theory. Approximatively 10% of observed main sequence (MS) and pre-main-sequence (PMS) radiative stars exhibit surface magnetic fields above the detection limit, raising the question of their origin. These stars host outer radiative envelopes, which are stably stratified. Therefore they...

We investigate free hydromagnetic eigenmodes of an incompressible, inviscid and ideal electrically conducting fluid in rotating triaxial ellipsoids. The container rotates with an angular velocity tilted from its figure. The magnetic base state is a uniform current density also tilted. Three-dimensional perturbations upon the base state are expanded...

We investigate free hydromagnetic eigenmodes of an incompressible, inviscid and ideal electrically conducting fluid in rotating triaxial ellipsoids. The container rotates with an angular velocity tilted from its figure. The magnetic base state is a uniform current density also tilted. Three-dimensional perturbations upon the base state are expanded...

Planetary cores consist of liquid metals (low Prandtl number $Pr$) that convect as the core cools. The convecting, conductive medium can self-excite and maintain a planetary magnetic field. Here we study nonlinear convection in a rotating (low, Ekman number $Ek$) planetary core using a fully 3D direct numerical simulation. Near the critical thermal...

The construction of geomagnetic, archaeomagnetic or palaeomagnetic field models requires some prior knowledge about the actual field, which can be gathered from the statistical properties of the field over a variety of length-scales and timescales. However, available geomagnetic data on centennial to millennial periods are too sparse to infer direc...

The Derviche Tourneur Sodium experiment, a spherical Couette magnetohydrodynamics experi- ment with liquid sodium as the medium and a dipole magnetic field imposed from the inner sphere, recently underwent upgrades to its diagnostics to better characterize the flow and induced magnetic fields with global rotation. In tandem with the upgrades, a set...

Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earth's core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in t...

Flows in natural systems are usually turbulent. The core of the Earth makes no exception. However, turbulence in the core departs from classical hydrodynamic turbulence because of the presence of a strong magnetic field and rotation. In this chapter, we work out how these two ingredients alter the organization of turbulence, and build plausible sce...

This corrects the article DOI: 10.1103/PhysRevLett.113.184501.

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 spherical harmonic transform has come from...

We test the ability of velocity fields inferred from geomagnetic secular
variation data to produce the global magnetic field of the Earth. Our kinematic
dynamo calculations use quasi-geostrophic (QG) flows inverted from geomagnetic
field models which, as such, incorporate flow structures that are Earth-like
and may be important for the geodynamo. F...

Seismic waves sensitive to the outermost part of the Earth's liquid core seem
to be affected by a stably stratified layer at the core-mantle boundary. Such a
layer could have an observable signature in both long-term and short-term
variations of the magnetic field of the Earth, which are used to probe the flow
at the top of the core. Indeed, with t...

The contribution of small scale turbulent fluctuations to the induction of
mean magnetic field is investigated in our liquid sodium spherical Couette
experiment with an imposed magnetic field.An inversion technique is applied to
a large number of measurements at $Rm \approx 100$ to obtain radial profiles of
the $\alpha$ and $\beta$ effects and maps...

We present a reconstruction of the mean axisymmetric azimuthal and meridional
flows in the DTS liquid sodium experiment. The experimental device sets a
spherical Couette flow enclosed between two concentric spherical shells where
the inner sphere holds a strong dipolar magnet, which acts as a magnetic
propeller when rotated. Measurements of the mea...

We use flows that we invert from two geomagnetic field models spanning
centennial time periods (gufm1 and COV-OBS), and apply Principal Component
Analysis and Singular Value Decomposition of coupled fields to extract the main
modes characterizing their spatial and temporal variations. The quasi
geostrophic flows inverted from both geomagnetic field...

Convection in planetary cores can generate fluid flow and magnetic fields, and a number of sophisticated codes exist to simulate the dynamic behaviour of such systems. We report on the first community activity to compare numerical results of computer codes designed to calculate fluid flow within a whole sphere. The flows are incompressible and rapi...

Metallic diapirs may have strongly contributed to core formations during the first million years of planetary evolutions. The aim of this study is to determine whether the dynamics induced by the diapir sinking can drive a dynamo and to characterize the required conditions on the size of the diapir, the mantle viscosity and the planetary latitude a...

Several teams have reported peculiar frequency spectra for flows in a
spherical shell. To address their origin, we perform numerical simulations of
the spherical Couette flow in a dipolar magnetic field, in the configuration of
the DTS experiment. The frequency spectra computed from time-series of the
induced magnetic field display similar bumpy sp...

Data assimilation aims at producing an optimal estimate of the state of the dy-
namical system one is interested in by combining two sources of information:
physical laws (in the form of a numerical model) and observations. A manda-
tory step during the development of a data assimilation framework involves a
validation phase using synthetic data. I...

Alfvén waves propagate in electrically conducting fluids in the presence of a magnetic field. Their reflection properties depend on the ratio between the kinematic viscosity and the magnetic diffusivity of the fluid, also known as the magnetic Prandtl number Pm. In the special case, Pm = 1, there is no reflection on an insulating, no-slip boundary,...

A dynamo mechanism explains the dipolar magnetic field of Jupiter and
the multipolar magnetic field of Neptune in terms of the width of the
zonal jet streams observed at their surfaces.

We present arguments supporting the hypothesis that the ow in the Earth's core, for the time scales of the historical secular variation, is well described by a quasi-geostrophic (QG) model, almost invariant along the rotation axis. A previous study showed that for axisymmetric motions, the dimensionless number appropriate to compare magnetic and ro...

We present a dynamo mechanism arising from the presence of barotropically
unstable zonal jet currents in a rotating spherical shell. The shear
instability of the zonal flow develops in the form of a global Rossby mode,
whose azimuthal wavenumber depends on the width of the zonal jets. We obtain
self-sustained magnetic fields at magnetic Reynolds nu...

Quasi-geostrophic (QG) flows are a recently developed and very promising paradigm for modeling decadal secular variation (SV). Here we examine the effects of allowing anisotropy and departures of the flow from quasigeostrophy. We perform dedicated numerical experiments of the flow dynamics and magnetic induction inside the Earth’s liquid core at ti...

We analyse by numerical simulations the nonlinear dynamics of the elliptic instability in the configurations of a single strained vortex and a system of two counter-rotating vortices. We show that although a weakly nonlinear regime associated with a limit cycle is possible, the nonlinear evolution far from the instability threshold is, in general,...

Hollow current density profiles often exist in Tokamak discharges, either transiently in the current ramp up, or in stationary phases where their main interest is to be generally associated with improved core confinement. Such a configuration is particularly relevant for Steady-State non-inductive scenarios where the self-driven bootstrap current d...

The three-dimensional linear temporal stability properties of a flow composed of two corotating q-vortices also called Batchelor vortices are predicted by numerical stability analysis. As for the corresponding counter-rotating case, when the axial flow parameter is increased, different instability modes are observed and identified as a combination...

Technical report TR1.2.1-3 du contrat européen FAR-Wake

We study the nonlinear evolution of the elliptic instability and its influence on the merging process of two corotating
Batchelor vortices using a spectral DNS approach.
First, we analyse the nonlinear saturation of the elliptic instability for a single strained vortex, with and without
axial jet, for moderate Reynolds numbers (Re = Γ/ν ≈ 12500, wh...

Rapidly rotating spherical kinematic dynamos at very low Ekman and Prandtl numbers are computed using the combination of a quasi-geostrophic (QG) model for the velocity field and a classical spectral 3D code for the magnetic field. The QG flow is computed in the equatorial plane of the sphere; it corresponds to Rossby wave instabilities of a geostr...

http://www2.mech.kth.se/efmc6/

We use a quasi-geostrophic numerical model to study the turbulence of rotating flows in a sphere, with realistic Ekman friction and bulk viscous dissipation. The forcing is caused by the destabilization of an axisymmetric Stewartson shear layer, generated by differential rotation, resulting in a forcing at rather large scales.
The equilibrium regi...

Detached shear layers in a rotating container, known as Stewartson layers, become unstable for a critical shear measured by the Rossby number Ro. To study rapidly rotating flows at asymptotically small Rossby and Ekman numbers E, a quasigeostrophic (QG) model is developed, whose main original feature is to enforce mass conservation properly, and va...

By averaging the Navier-Stokes equation along the rotation axis, we developed an enhanced Quasi-Geostrophic (QG) model that is able to handle barotropic rotating fluid flows in an axisymmetric container with finite slopes. This QG-model is used to study the instabilities of a shear layer known as the Stewartson layer. We show that the slope is the...

We study the destabilization of a shear layer, produced by differential rotation of a rotating axisymmetric container. For small forcing, this produces a shear layer, which has been studied by Stewartson and is almost invariant along the rotation axis. When the forcing increases, instabilities develop. To study the asymptotic regime (very low Ekman...

The frequency of plume formation in fully-developed thermal convection is determined experimentally. Because the fluid has a temperature-dependent viscosity, the cold and hot thermal boundary layers have different thick-nesses and viscosities. As a result, plumes are released from these layers with different frequencies. There also appears to be a...

La diffusion d’ultrasons par un ́ecoulement permet de mesurer la vorticit ́e,
en s ́electionnant une ́echelle de l’ ́ecoulement. Par effet Doppler, on acc`ede
́egalement `a la distribution de vitesse.
Nous avons dans un premier temps quantifi ́e et pr ́ecis ́e la mesure de la
vitesse et du taux de turbulence, en tenant compte des effets de diffract...

Taking advantage of the properties of liquid metals and of rapidly rotating flows, we are able to compute dynamos at high Reynolds number (£ ¥ ¤ ¥ ©) and low magnetic Prandtl number (¥ ©) We developed a numerical model that uses a quasi-goestrophic approximation to compute the flow (without subgrid scale model), leading to two-dimensional equations...

## Projects

Projects (4)

Evaluate the application of new time integration targeting climate and weather simulations. This involves e.g. studying its mathematical as well as high-performance computing aspects.