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Journal Article•DOI•

Turbulent geodynamo simulations: a leap towards Earth's core

Nathanaël Schaeffer1, Dominique Jault1, Henri-Claude Nataf1, Alexandre Fournier2•
University of Grenoble1, Institut de Physique du Globe de Paris2
01 Oct 2017-Geophysical Journal International (Oxford Academic)-Vol. 211, Iss: 1, pp 1-29
TL;DR: In this paper, a sequence of three convection-driven simulations in a rapidly rotating spherical shell is used to reach realistic turbulent regime in direct numerical simulations of the geodynamo.
Abstract: 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 vigorous convection (magnetic Reynolds number Rm>500) and rapid rotation (Ekman number E=1e-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 Alfven 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, large-scale 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 MAC balance – involving Coriolis, Lorentz and buoyancy forces. However, small-scale (E^{1/3}) quasi-geostrophic convection is still observed in the regions of low magnetic intensity.
Citations
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Journal Article•DOI•
Dynamo theories.

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Francois Rincon
15 Mar 2019-arXiv: Plasma Physics
TL;DR: In this paper, the authors provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field.
Abstract: These lecture notes are based on a tutorial given in 2017 at a plasma physics winter school in Les Houches. Their aim is to provide a self-contained graduate-student level introduction to the theory and modelling of the dynamo effect in turbulent fluids and plasmas, blended with a review of current research in the field. The primary focus is on the physical and mathematical concepts underlying different (turbulent) branches of dynamo theory, with some astrophysical, geophysical and experimental context disseminated throughout the document. The text begins with an introduction to the rationale, observational and historical roots of the subject, and to the basic concepts of magnetohydrodynamics relevant to dynamo theory. The next two sections discuss the fundamental phenomenological and mathematical aspects of (linear and nonlinear) small- and large-scale MHD dynamos. These sections are complemented by an overview of a selection of current active research topics in the field, including the numerical modelling of the geo- and solar dynamos, shear dynamos driven by turbulence with zero net helicity, and MHD-instability-driven dynamos such as the magnetorotational dynamo. The difficult problem of a unified, self-consistent statistical treatment of small and large-scale dynamos at large magnetic Reynolds numbers is also discussed throughout the text. Finally, an excursion is made into the relatively new but increasingly popular realm of magnetic-field generation in weakly-collisional plasmas. A short discussion of the outlook and challenges for the future of the field concludes the presentation.

111 citations

Cites background or methods from "Turbulent geodynamo simulations: a ..."

  • ...…the problem (Christensen et al. 1999; Olson & Christensen 2006; Christensen & Aubert 2006; Takahashi et al. 2008; Soderlund et al. 2012; Schrinner et al. 2012; Stelzer & Jackson 2013; Dormy 2016; Sheyko et al. 2016; Yadav et al. 2016; Aubert et al. 2017; Schaeffer et al. 2017; Sheyko et al. 2018)....

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  • ...A long-time strategy of the geodynamo community to address this question has been to carry out a methodical parametric numerical exploration of dynamical force balances, scaling laws, and nonlinear dynamo states along parameter paths consistent with the natural ordering of scales in the problem (Christensen et al. 1999; Olson & Christensen 2006; Christensen & Aubert 2006; Takahashi et al. 2008; Soderlund et al. 2012; Schrinner et al. 2012; Stelzer & Jackson 2013; Dormy 2016; Sheyko et al. 2016; Yadav et al. 2016; Aubert et al. 2017; Schaeffer et al. 2017; Sheyko et al. 2018)....

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  • ...05 and Rm = O(10(3)), on the other hand, now seems on the verge of convergence towards asymptotic strong-field magnetostrophic dynamo states (Yadav et al. 2016; Aubert et al. 2017; Schaeffer et al. 2017; Sheyko et al. 2018)....

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  • ...s (Schekochihin et al. 2002c). Yet another possibility, explored by Schekochihin et al. (2004b), is to postulate a local magnetic-field-orientation dependent anisotropic correction κ to the correlation tensor κ of the Kazantsev velocity field to model the effects of magnetic tension on the flow....

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  • ...…simulations of the geodynamo extending down to E = O(10−7), Pm = 0.05 and Rm = O(103), on the other hand, now seems on the verge of convergence towards asymptotic strong-field magnetostrophic dynamo states (Yadav et al. 2016; Aubert et al. 2017; Schaeffer et al. 2017; Sheyko et al. 2018)....

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Journal Article•DOI•
Geomagnetic jerks and rapid hydromagnetic waves focusing at Earth's core surface

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Julien Aubert1, Christopher C. Finlay2•
Institut de Physique du Globe de Paris1, Technical University of Denmark2
01 May 2019-Nature Geoscience
TL;DR: In this paper, the authors show that the acceleration of Earth's magnetic field can be explained by the arrival of localized Alfven wave packets radiated from sudden buoyancy releases inside the core.
Abstract: Geomagnetic jerks are abrupt changes in the second time derivative—the secular acceleration—of Earth’s magnetic field that punctuate ground observatory records. As their dynamical origin has not yet been established, they represent a major obstacle to the prediction of geomagnetic field behaviour for years to decades ahead. Recent jerks have been linked to short-lived, temporally alternating and equatorially localized pulses of secular acceleration observed in satellite data, associated with rapidly alternating flows at Earth’s core surface. Here we show that these signatures can be reproduced in numerical simulations of the geodynamo that realistically account for the interaction between slow core convection and rapid hydromagnetic waves. In these simulations, jerks are caused by the arrival of localized Alfven wave packets radiated from sudden buoyancy releases inside the core. As they reach the core surface, the waves focus their energy towards the equatorial plane and along lines of strong magnetic flux, creating sharp interannual changes in core flow and producing geomagnetic jerks through the induced variations in magnetic field acceleration. The ability to numerically reproduce jerks offers a new way to probe the physical properties of Earth’s deep interior. Geomagnetic jerks in the Earth’s magnetic field are caused by the arrival of hydromagnetic waves and could be generated by sudden releases of buoyancy in the Earth’s core, suggest geodynamic numerical model simulations.

75 citations

Journal Article•DOI•
Turbulent convective length scale in planetary cores

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Céline Guervilly1, Philippe Cardin2, Nathanaël Schaeffer2•
Newcastle University1, University of Savoy2
01 Jun 2019-Nature
TL;DR: Numerical modelling of rotating turbulent convective flows shows that the length scale of convection in planetary cores is set by the flow speed and not by the fluid viscosity, and the need to resolve the numerically inaccessible viscous scale could be relaxed in future more realistic geodynamo simulations.
Abstract: 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 strong dependence of these properties on planetary rotation, buoyancy driving and magnetic fields, all of which are difficult to model using realistic conditions. In the absence of strong magnetic fields, the convective flows of the core are expected to be in a regime of rapidly rotating turbulence1, which remains largely unexplored. Here we use a combination of non-magnetic numerical models designed to explore this regime to show that the convective length scale becomes independent of the viscosity when realistic parameter values are approached and is entirely determined by the flow velocity and the planetary rotation. The velocity decreases very rapidly at smaller scales, so this turbulent convective length scale is a lower limit for the energy-carrying length scales in the flow. Using this approach, we can model realistically the dynamics of small non-magnetic cores such as the Moon. Although modelling the conditions of larger planetary cores remains out of reach, the fact that the turbulent convective length scale is independent of the viscosity allows a reliable extrapolation to these objects. For the Earth’s core conditions, we find that the turbulent convective length scale in the absence of magnetic fields would be about 30 kilometres, which is orders of magnitude larger than the ten-metre viscous length scale. The need to resolve the numerically inaccessible viscous scale could therefore be relaxed in future more realistic geodynamo simulations, at least in weakly magnetized regions. Numerical modelling of rotating turbulent convective flows shows that the length scale of convection in planetary cores is set by the flow speed and not by the fluid viscosity.

66 citations

Journal Article•DOI•
Rotating thermal convection in liquid gallium: multi-modal flow, absent steady columns

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Jonathan M. Aurnou1, Vincent Bertin2, Vincent Bertin1, A. M. Grannan1, Susanne Horn1, Tobias Vogt3 •
University of California, Los Angeles1, École Normale Supérieure2, Helmholtz-Zentrum Dresden-Rossendorf3
10 Jul 2018-Journal of Fluid Mechanics
TL;DR: In this paper, a cylinder of aspect ratio with liquid gallium as the working fluid was used to study the dynamics of low-number convective flows in low-volatile fluids.
Abstract: Earth’s magnetic field is generated by convective motions in its liquid metal core. In this fluid, the heat diffuses significantly more than momentum, and thus the Prandtl number is well below unity. The thermally driven convective flow dynamics of liquid metals are very different from moderate- fluids, such as water and those used in current dynamo simulations. In order to characterise rapidly rotating thermal convection in low- number fluids, we have performed laboratory experiments in a cylinder of aspect ratio using liquid gallium ( ) as the working fluid. The Ekman number varies from to and the Rayleigh number varies from to . Using spectral analysis stemming from point-wise temperature measurements within the fluid and measurements of the Nusselt number , we characterise the different styles of low- rotating convective flow. The convection threshold is first overcome in the form of container-scale inertial oscillatory modes. At stronger forcing, sidewall-attached modes are identified for the first time in liquid metal laboratory experiments. These wall modes coexist with the bulk oscillatory modes. At well below the values where steady rotating columnar convection occurs, the bulk flow becomes turbulent. Our results imply that rotating convective flows in liquid metals do not develop in the form of quasisteady columns, as in moderate- fluids, but in the form of oscillatory convective motions. Thus, thermally driven flows in low- geophysical and astrophysical fluids can differ substantively from those occurring in models. Furthermore, our experimental results show that relatively low-frequency wall modes are an essential dynamical component of rapidly rotating convection in liquid metals.

56 citations

Cites background from "Turbulent geodynamo simulations: a ..."

  • ...In Earth’s core, E ' 10−15, whereas E & 10−7 in present-day direct numerical simulations of core processes (e.g. Stellmach et al. 2014; Schaeffer et al. 2017)....

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  • ...This point seems especially relevant, given the outsized influence that tangent cylinder flows seem to exert in global-scale dynamo generation processes and on the magnetic field morphology in a variety of models (e.g. Glatzmaier & Roberts 1995; Aubert et al. 2008; Schaeffer et al. 2017)....

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  • ...These differences persist in fully turbulent convection cases where tangent cylinder flows appear to be dominant contributors to the global dynamo action, both in the earliest dynamo models and in the highest-resolution models carried out to date (Glatzmaier & Roberts 1995; Schaeffer et al. 2017)....

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  • ...…columnar convection modes that dominate models of Pr & 1 rotating convection (e.g. Grooms et al. 2010; King & Aurnou 2012; Gastine, Wicht & Aubert 2016) and dynamo action (e.g. Christensen 2011; Jones 2011; Schaeffer et al. 2017) are non-existent in the experimental simulations carried out here....

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  • ...…dynamo action (e.g. Calkins et al. 2015; Nataf & Schaeffer 2015; Calkins 2017), as likely occurs in a number of solar system planetary dynamos and in the majority of present-day numerical dynamo models (e.g. Soderlund et al. 2015; Yadav et al. 2016a; Aurnou & King 2017; Schaeffer et al. 2017)....

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Journal Article•DOI•
Assimilation of ground and satellite magnetic measurements: inference of core surface magnetic and velocity field changes

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Olivier Barrois1, Magnus Danel Hammer2, Christopher C. Finlay2, Y Martin1, Nicolas Gillet1 •
University of Grenoble1, Technical University of Denmark2
01 Oct 2018-Geophysical Journal International

51 citations

Cites background from "Turbulent geodynamo simulations: a ..."

  • ...This is a common configuration for polar vortices found in the most up to date numerical simulations (Schaeffer et al. 2017), which show much variability through epochs....

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References
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Book•
Magnetic field generation in electrically conducting fluids

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H. K. Moffatt
01 Jan 1978

2,987 citations

Journal Article•DOI•
A three-dimensional self-consistent computer simulation of a geomagnetic field reversal

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Gary A. Glatzmaiers1, Paul H. Roberts2•
Los Alamos National Laboratory1, University of California, Los Angeles2
01 Sep 1995-Nature
TL;DR: In this article, a three-dimensional, self-consistent numerical model of the geodynamo is described, which maintains a magnetic field for over 40,000 years, including a successful reversal of the dipole moment.
Abstract: A three-dimensional, self-consistent numerical model of the geodynamo is described, that maintains a magnetic field for over 40,000 years. The model, which incorporates a finitely conducting inner core, undergoes several polarity excursions and then, near the end of the simulation, a successful reversal of the dipole moment. This simulated magnetic field reversal shares some features with real reversals of the geomagnetic field, and may provide insight into the geomagnetic reversal mechanism.

779 citations

Journal Article•DOI•
Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields

[...]

Ulrich R. Christensen1, Julien Aubert2•
Max Planck Society1, Institut de Physique du Globe de Paris2
01 Jul 2006-Geophysical Journal International
TL;DR: In this paper, an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude, were studied and their scaling laws were established.
Abstract: SUMMARY We study numerically an extensive set of dynamo models in rotating spherical shells, varying all relevant control parameters by at least two orders of magnitude. Convection is driven by a fixed temperature contrast between rigid boundaries. There are two distinct classes of solutions with strong and weak dipole contributions to the magnetic field, respectively. Non-dipolar dynamos are found when inertia plays a significant role in the force balance. In the dipolar regime the critical magnetic Reynolds number for self-sustained dynamos is of order 50, independent of the magnetic Prandtl number Pm. However, dynamos at low Pm exist only at sufficiently low Ekman number E. For dynamos in the dipolar regime we attempt to establish scaling laws that fit our numerical results. Assuming that diffusive effects do not play a primary role, we introduce non-dimensional parameters that are independent of any diffusivity. These are a modified Rayleigh number based on heat (or buoyancy) flux Ra ∗ , the Rossby number Ro measuring the flow velocity, the Lorentz number Lo measuring magnetic field strength, and a modified Nusselt number Nu ∗ for the advected heat flow. To first approximation, all our dynamo results can be collapsed into simple power-law dependencies on the modified Rayleigh number, with approximate exponents of 2/5, 1/2 and 1/3 for the Rossby number, modified Nusselt number and Lorentz number, respectively. Residual dependencies on the parameters related to diffusion (E, Pm, Prandtl number Pr) are weak. Our scaling laws are in agreement with the assumption that the magnetic field strength is controlled by the available power and not necessarily by a force balance. The Elsasser number � , which is the conventional measure for the ratio of Lorentz force to Coriolis force, is found to vary widely. We try to assess the relative importance of the various forces by studying sources and sinks of enstrophy (squared vorticity). In general Coriolis and buoyancy forces are of the same order, inertia and viscous forces make smaller and variable contributions, and the Lorentz force is highly variable. Ignoring a possible weak dependence on the Prandtl numbers or the Ekman number, a surprising prediction is that the magnetic field strength is independent both of conductivity and of rotation rate and is basically controlled by the buoyancy flux. Estimating the buoyancy flux in the Earth’s core using our Rossby number scaling and a typical velocity inferred from geomagnetic secular variations, we predict a small growth rate and old age of the inner core and obtain a reasonable magnetic field strength of order 1 mT inside the core. From the observed heat flow in Jupiter, we predict an internal field of 8 mT, in agreement with Jupiter’s external field being 10 times stronger than that of the Earth.

719 citations

"Turbulent geodynamo simulations: a ..." refers background or methods or result in this paper

  • ...However, the magnetic energy of all three simulations correspond to the trend exhibited by the Christensen and Aubert (2006) dataset, which also displays a scatter by a factor 3 to 10 (not shown)....

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  • ...Reanalyzing the large suite of simulations of Christensen and Aubert (2006), King and Buffett (2013) confirmed that viscous dissipation was far from negligible in these simulations; Cheng and Aurnou (2016) showed that the diffusionless scaling laws were hiding an actual dependency upon viscosity;…...

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  • ...Following (Christensen and Aubert, 2006), the fraction of axial dipole in the observable spectrum (up to ` = 13) is given by fdip in table 2, and is in reasonable agreement with the Earth’s value (fdip ' 0....

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  • ...Those initial conditions are obtained by applying previously established scaling laws (Christensen and Aubert, 2006) to the output of a lower resolution simulation at parameters further from the Earth’s core....

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  • ...Christensen and Aubert (2006) introduced an average harmonic degree ¯̀ = ∑ ` `E`(u)/ ∑ `E`(u)....

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Journal Article•DOI•
Equations governing convection in earth's core and the geodynamo

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Stanislav I. Braginsky1, Paul H. Roberts1•
University of California, Los Angeles1
01 Aug 1995-Geophysical and Astrophysical Fluid Dynamics
TL;DR: In this article, a closed system of equations and boundary conditions is derived that governs core convection and the geodynamo, and it is concluded that compositional convection may not dominate thermal convection, as had previously been argued by Braginsky.
Abstract: Convection in Earth's fluid core is regarded as a small deviation from a well-mixed adiabatic state of uniform chemical composition. The core is modeled as a binary alloy of iron and some lighter constituent, whose precise chemical composition is unknown but which is here assumed to be FeAd, where Ad = Si, O or S. The turbulent transport of heat and light constituent is considered, and a simple ansatz is proposed in which this is modeled by anisotropic diffusion. On this basis, a closed system of equations and boundary conditions is derived that governs core convection and the geodynamo. The dual (thermal + compositional) nature of core convection is reconsidered. It is concluded that compositional convection may not dominate thermal convection, as had previously been argued by Braginsky (Soviet Phys. Dokl., v. 149, p. 8, 1963; Geomag, and Aeron., v. 4, p. 698, 1964), but that the two mechanisms are most probably comparable in importance. The key parameters leading to this conclusion are isolated...

483 citations

"Turbulent geodynamo simulations: a ..." refers methods in this paper

  • ...We use the codensity formulation (Braginsky and Roberts, 1995), which combines buoyancy effects due to temperature variations and chemical species concentration into one scalar codensity field C, with associated diffusivity κ....

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Journal Article•DOI•
On almost rigid rotations

[...]

K. Stewartson1•
University of Bristol1
01 Oct 1957-Journal of Fluid Mechanics
TL;DR: In this paper, it was shown that if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layers on the other at the same value of r.
Abstract: In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

458 citations

"Turbulent geodynamo simulations: a ..." refers background in this paper

  • ...A sharp shear layer, associated with a meridional circulation materializes the tangent cylinder, and is reminiscent of Stewartson layers (Stewartson, 1966)....

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