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Turbulent convective length scale in planetary cores

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.

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Fig. 1: Flow in the 3D model.
Fig. 2: Effect of the Rossby number on the flow structure.
Fig. 3: Distribution of the kinetic energy at different length scales.
Fig. 4: Scaling of the velocity and length scale.

Data availability

Source data for Figs. 3, 4 are provided with this paper. The data generated during this study are included in the Supplementary Information file. Any additional data that support the findings of this study are available from the corresponding author on reasonable request.

Code availability

The 3D numerical code XSHELLS is freely available at https://bitbucket.org/nschaeff/xshells and is distributed under the open source CeCILL License (http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.html). The QG numerical code is available from the corresponding author on request.

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Acknowledgements

C.G. was supported by the UK Natural Environment Research Council under grant NE/M017893/1. P.C. and N.S. were supported by the French Agence Nationale de la Recherche under grants ANR-13-BS06-0010 (TuDy) and ANR-14-CE33-0012 (MagLune). N.S. acknowledges GENCI for access to the Occigen resource (CINES) under grants A0020407382 and A0040407382. This research made use of the Rocket High Performance Computing service at Newcastle University, the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk), and the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (http://www.dirac.ac.uk) and funded by BEIS capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, and Durham University and STFC operations grant ST/R000832/1. Some computations were also performed on the Froggy platform of CIMENT (https://ciment.ujf-grenoble.fr), supported by the Rhône-Alpes region (CPER07_13 CIRA), OSUG@2020 LabEx (ANR10 LABX56) and Equip@Meso (ANR10 EQPX-29-01). ISTerre is part of Labex OSUG@2020 (ANR10 LABX56).

Reviewer information

Nature thanks Bruce Buffett and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations

Authors

Contributions

C.G. and P.C. performed the numerical simulations with the QG code. N.S. performed the numerical simulations with the 3D code. All authors contributed to the analysis of the data and the preparation of the manuscript.

Corresponding author

Correspondence to Céline Guervilly.

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Extended data figures and tables

Extended Data Fig. 1 Scaling of the Reynolds number.

Re as a function of Ra × Ek/Pr in simulations performed with the 3D model (green data points) for Ek [10−8, 10−6], the QG model (blue data points) for Ek [10−11, 10−6], and the hybrid model (red data points) for Ek [10−8, 10−7]. Marker colours correspond to Ekman numbers (values given in the key) and marker shapes correspond to Prandtl numbers (circles, Pr = 10−2 and squares, Pr = 10−1). The dashed line represents Re = 0.6Ra × Ek/Pr. Inset, the same data compensated by theoretical scaling as a function of Ra × Ek/Pr.

Extended Data Fig. 2 Comparison of the radial length scale with the azimuthal length scale.

Radial scale of the convective flows \({{\mathscr{L}}}_{{\rm{r}}}(s)\) as a function of the azimuthal length scale \({\mathscr{L}}(s)\) obtained with the QG model at different radii s. Marker colours correspond to Ekman numbers (with Pr = 10−2) and marker shapes correspond to the given radii. The radial scale is calculated from auto-correlation functions of the radial velocity, and the convective length scale corresponds to an azimuthal scale calculated from the peak of the power spectra of the radial kinetic energy at radius s. The dashed line represents \({{\mathscr{L}}}_{{\rm{r}}}(s)={\mathscr{L}}(s)\).

Extended Data Fig. 3 Variation of the convective length scale with radius.

Convective length scale \({\mathscr{L}}(s)\) as a function of Ro(s)/|β| obtained with the QG model at different radii s. Marker colours correspond to Ekman numbers, solid-colour markers correspond to Pr = 0.01, dotted markers to Pr = 0.1, and marker shapes correspond to the given radii. The convective length scale corresponds to an azimuthal scale calculated from the peak of the power spectra of the radial kinetic energy at radius s. The dashed line represents \({{\mathscr{L}}}_{{\rm{r}}}(s)=6{\left({\rm{Ro}}(s)/| \beta | \right)}^{1/2}\). Inset, the length scale compensated by theoretical scaling as a function of Ro(s)/|β|.

Extended Data Fig. 4 Effect of the heating mode on the convective length scale.

Convective length scale \({\mathscr{L}}\) as a function of Ro obtained with the QG model for internal heating (IH, same points as in Fig. 4) and differential heating (DH) with an inner core of radius Ri = 0.35. Ek [10−11, 10−6] and Pr {10−2, 10−1, 1} are given in the key. The convective scale is averaged over radii between s = 0.1 and 0.6 and the vertical error bars give the standard deviation in this interval. The dashed line represents \({\mathscr{L}}=11{{\rm{Ro}}}^{1/2}\). Inset, the same data compensated by theoretical scaling as a function of Ro.

Extended Data Fig. 5 Time series of the kinetic energy density for two representative simulations.

a, b, Time series of the kinetic energy density K and the kinetic energy density of the axisymmetric flow Kaxi for Ek = 10−11, Pr = 0.01 and Ra = 3.75 × 1013 using the QG model (a) and Ek = 10−8, Pr = 0.01 and Ra = 2 × 1010 using the 3D model (b). Time is given in units of a viscous timescale.

Supplementary information

Supplementary Table

This file contains details of the numerical simulations. It shows a list of input and output parameters for the simulations performed with the 3D and QG models.

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Guervilly, C., Cardin, P. & Schaeffer, N. Turbulent convective length scale in planetary cores. Nature 570, 368–371 (2019). https://doi.org/10.1038/s41586-019-1301-5

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