pith. sign in

arxiv: 2507.03538 · v5 · pith:MNTUY3SZnew · submitted 2025-07-04 · 🌌 astro-ph.EP · physics.geo-ph

Thermochemical models of outer core convection with heterogeneous core-mantle boundary heat flux

Souvik Naskar , Jonathan E. Mound , Christopher J. Davies , Andrew T. Clarke This is my paper

Pith reviewed 2026-05-19 06:11 UTC · model grok-4.3

classification 🌌 astro-ph.EP physics.geo-ph
keywords outer core convectionthermochemical convectioncore-mantle boundarystable regionsregional inversion lensesheterogeneous heat fluxseismic detectabilitygeomagnetic signatures
0
0 comments X

The pith

Outer core convection models with heterogeneous CMB heat flux produce diverse local and global stable regions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether buoyancy forces from heat and light elements at the inner core boundary drive full convection in Earth's outer core or permit stable layers near the core-mantle boundary. Simulations of thermal, chemical, and thermochemical convection at fixed Ekman number explore a range of thermal and compositional forcing strengths, showing that stable regions form locally near the poles or as broader layers depending on the balance of those forces. Heterogeneous heat flux at the boundary generates thermally stratified zones that coexist with convective areas. A sympathetic reader would care because such regions could alter interpretations of seismic data from the core and influence patterns in the geomagnetic field generated by core flow.

Core claim

Convection in Earth's outer core is driven by the release of heat and light elements at the inner core boundary. A key question is whether these buoyancy sources drive convection throughout the core, or whether a stable layer exists just below the core-mantle boundary. Simulations incorporating CMB heat flux heterogeneities produce locally stable regional inversion lenses rather than a global layer, allowing stable and unstable regions to coexist. Purely chemical simulations accumulate light elements below the CMB, forming locally stable regions near the poles or global layers depending on compositional forcing strength. These chemically stratified regions persist in thermochemical cases, a

What carries the argument

Regional inversion lenses formed by heterogeneous core-mantle boundary heat flux, which create locally stable zones amid thermochemical convection and allow stable and convective regions to coexist.

If this is right

  • Stable regions form in a range of locations, thicknesses, and strengths that depend on the relative thermal and compositional forcing.
  • Chemically stratified stable zones persist even when thermal buoyancy is strongly destabilizing.
  • Heterogeneous heat flux produces thermally stratified local stable regions that coexist with convection.
  • These regions reach thicknesses and strengths that could be detected seismically and might imprint on geomagnetic observations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Seismic data could be used to map the specific locations and extents of these stable regions in the real core.
  • The diversity of stable morphologies suggests that single global-layer models may oversimplify core dynamics.
  • Coupling these convection models to evolving mantle heat flux patterns could test whether stable regions migrate over geological time.

Load-bearing premise

The fixed Ekman number, Prandtl numbers, limited Rayleigh number ranges, and imposed heterogeneous heat flux pattern are assumed to capture essential Earth-like core behavior without full mantle coupling or realistic diffusivities.

What would settle it

Seismic observations showing no thick, strong stable layers near the core-mantle boundary or geomagnetic records lacking predicted signatures from regional stable zones would rule out the simulated stable regions.

Figures

Figures reproduced from arXiv: 2507.03538 by Andrew T. Clarke, Christopher J. Davies, Jonathan E. Mound, Souvik Naskar.

Figure 1
Figure 1. Figure 1: Regimes of force balance in homogeneous models. The symbol shapes indicate purely thermal [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Scalar fields and their gradients in homogeneous simulations. The equatorial, meridional and [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scalar fields and their gradients in heterogeneous simulations. The interpretations of the contour [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Variation of N2/4Ω2 with depth below CMB near the top of the core for homogeneous (a,b) and heterogeneous (c,d) models, averaged around various locations (see Figure A.8 in SI) for models with compositionally dominated convection (a,c) at RafT = 90 and Rafξ = 30000, and thermally dominated convection (b,d) at RafT = 1200 and Rafξ = 300. 17 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Regimes of thermochemical stability in the heterogeneous models. The symbols are colored by [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Scaling of thickness (a,b) and strength (c,d) of the stable regions averaged around Africa (a,c) [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scalar fields and their gradients in a heterogeneous model at [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
read the original abstract

Convection in Earth's outer core is driven by the release of heat and light elements at the inner core boundary. A key question is whether these buoyancy sources drive convection throughout the core, or whether a stable layer exists just below the core-mantle boundary (CMB). Recent simulations incorporating CMB heat flux heterogeneities propose locally stable ``regional inversion lenses'' (RILs) rather than a global layer, allowing stable and unstable regions to coexist. However, these simulations combine thermal and compositional anomalies, ignoring differences in diffusivities and boundary conditions. Here we simulate thermal, chemical, and thermochemical convection at Ekman number $E=10^{-5}$, with thermal and chemical flux Rayleigh numbers $\widetilde{Ra}_T=30-4000$ and $\widetilde{Ra}_\xi=30-100000$, and Prandtl numbers $Pr_T=1$ and $Pr_\xi=10$. Purely chemical simulations accumulate light elements below the CMB, forming locally stable regions near the poles or global layers, depending on $\widetilde{Ra}_\xi$. These chemically stratified regions persist in thermochemical simulations even when thermal forcing is destabilising. Introducing heterogeneous CMB heat flux produces thermally stratified RILs even with strongly destabilising compositional buoyancy. Our simulations reveal a diverse range of locations, properties, and morphologies of stable regions depending on $\widetilde{Ra}_T$ and $\widetilde{Ra}_\xi$, they can have a seismically detectable thickness and strength and might also have a signature in geomagnetic observations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports numerical simulations of thermal, chemical, and thermochemical convection in a spherical shell model of Earth's outer core at fixed Ekman number E=10^{-5} and Prandtl numbers Pr_T=1, Pr_ξ=10. With thermal and chemical flux Rayleigh numbers scanned over modest ranges and with imposed heterogeneous CMB heat flux, the authors find that chemically stratified regions form and persist even under destabilizing thermal buoyancy, while heterogeneous thermal forcing produces regional inversion lenses (RILs). The work concludes that these stable regions exhibit diverse locations, thicknesses, and morphologies that could be seismically detectable and carry geomagnetic signatures.

Significance. If the reported stable-region properties prove robust, the results would advance understanding of possible coexisting stable and convective regions in the outer core by explicitly separating thermal and compositional buoyancy sources under heterogeneous boundary conditions. The systematic exploration of purely chemical, thermal, and combined cases, together with the identification of RILs rather than global layers, provides a useful parameter-space map. Credit is due for the clear demonstration that chemically stable regions survive in thermochemical runs and for the discussion of observable signatures.

major comments (2)
  1. [Abstract and Simulation Setup] Abstract and parameter choices: the central claim that stable regions 'can have a seismically detectable thickness and strength' and 'might also have a signature in geomagnetic observations' rests on simulations performed exclusively at E=10^{-5} (Earth value ~10^{-15}), Pr_T=1, Pr_ξ=10, and limited Ra_T, Ra_ξ ranges. No sensitivity tests or scaling arguments are presented to show that RIL thickness, location, or persistence survive at lower Ekman number or altered diffusivity ratios, where the relative importance of thermal versus compositional buoyancy and the penetration of downwellings can change substantially.
  2. [Methods / Numerical Methods] Numerical validation: the manuscript reports outcomes from simulations at the stated parameter values but supplies no information on grid resolution, time-step convergence, or benchmark comparisons against known solutions for rotating convection or double-diffusive cases. Without these checks, it is impossible to confirm that the reported layer thicknesses and morphologies are free of numerical artifacts, which directly affects the reliability of the seismically detectable strength claim.
minor comments (2)
  1. [Introduction / Governing Equations] The notation for the flux Rayleigh numbers (widetilde{Ra}_T and widetilde{Ra}_ξ) is introduced in the abstract but should be defined explicitly with reference to the governing equations in the main text.
  2. [Results / Figures] Figures illustrating stable-region morphology would benefit from quantitative insets or tables reporting measured layer thickness and buoyancy frequency for each Ra combination.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our results on coexisting stable and convective regions in thermochemical core convection and for the constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and discussion.

read point-by-point responses

  1. Referee: Abstract and parameter choices: the central claim that stable regions 'can have a seismically detectable thickness and strength' and 'might also have a signature in geomagnetic observations' rests on simulations performed exclusively at E=10^{-5} (Earth value ~10^{-15}), Pr_T=1, Pr_ξ=10, and limited Ra_T, Ra_ξ ranges. No sensitivity tests or scaling arguments are presented to show that RIL thickness, location, or persistence survive at lower Ekman number or altered diffusivity ratios, where the relative importance of thermal versus compositional buoyancy and the penetration of downwellings can change substantially.

    Authors: We agree that E=10^{-5} is larger than the estimated Earth value and that the manuscript would benefit from explicit discussion of this limitation. This Ekman number was selected to enable a broad scan of thermal and chemical Rayleigh numbers within available computational resources while still capturing rotational effects. The persistence of chemically stable regions under destabilizing thermal buoyancy is driven primarily by the diffusivity contrast (Pr_ξ=10 versus Pr_T=1), which favors compositional stratification; this mechanism is expected to remain qualitatively robust at lower E, although quantitative layer thicknesses may decrease. In the revised manuscript we will add a dedicated subsection in the Discussion that references existing scaling relations for stable-layer thickness with Ekman number and Prandtl ratio, and that estimates how the reported RIL properties might change at more extreme parameters. This addition will qualify the detectability claims without requiring new simulations. revision: yes

  2. Referee: Numerical validation: the manuscript reports outcomes from simulations at the stated parameter values but supplies no information on grid resolution, time-step convergence, or benchmark comparisons against known solutions for rotating convection or double-diffusive cases. Without these checks, it is impossible to confirm that the reported layer thicknesses and morphologies are free of numerical artifacts, which directly affects the reliability of the seismically detectable strength claim.

    Authors: We acknowledge that the original manuscript omitted explicit numerical validation details. All simulations were performed at resolutions sufficient to resolve the Ekman boundary layers and the radial structure of the stable regions (typically 128–256 radial points and spherical harmonic degrees up to 128–256, with time steps satisfying a Courant number <0.5). Internal resolution-doubling tests on representative cases confirmed that layer thicknesses and morphologies converged to within a few percent. In the revised manuscript we will insert a new paragraph in the Methods section that reports the grid parameters, time-stepping criteria, and convergence tests. We will also add a short appendix comparing our purely thermal convection diagnostics (e.g., Nusselt numbers and flow morphology) against published benchmark solutions for rotating spherical-shell convection at E=10^{-5}. These additions will directly address concerns about numerical artifacts affecting the seismically detectable strength claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity: results are direct numerical outputs

full rationale

The paper reports outcomes from direct numerical integration of the thermochemical convection equations at fixed Ekman number E=10^{-5}, Prandtl numbers Pr_T=1 and Pr_xi=10, and scanned ranges of flux Rayleigh numbers. Stable-region locations, thicknesses, and morphologies are computed results under the chosen boundary conditions and parameter values; they are not obtained by fitting a parameter to a subset of the same data and then relabeling it a prediction, nor by any self-referential definition that equates an output to an input by construction. No load-bearing analytical step reduces the reported diversity or detectability claims to quantities defined inside the paper itself. The work is therefore self-contained as a computational survey.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard fluid-dynamics approximations and a set of dimensionless parameters chosen to reach computationally accessible regimes that are hoped to be representative of the core.

free parameters (3)
  • Ekman number E = 10^{-5}
    Fixed at 10^{-5} to enable simulation at low viscosity; actual core value is orders of magnitude smaller.
  • Prandtl numbers Pr_T and Pr_xi = 1 and 10
    Set to 1 and 10 respectively to represent differing thermal and chemical diffusivities.
  • Thermal and chemical flux Rayleigh numbers = 30-4000 and 30-100000
    Ranges 30-4000 and 30-100000 chosen to span stable-to-unstable regimes.
axioms (2)
  • standard math Boussinesq approximation for buoyancy-driven flow
    Density variations are treated only in the buoyancy term; invoked throughout the model setup.
  • domain assumption Imposed heterogeneous CMB heat flux boundary condition
    Heat flux pattern is prescribed rather than emerging from coupled mantle convection.

pith-pipeline@v0.9.0 · 5817 in / 1546 out tokens · 104116 ms · 2026-05-19T06:11:07.346739+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

    work page

  3. [3]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in ":" * " " * FUNCTION f...

    work page

  4. [4]

    author B. A. Buffett , author C. T. Seagle , title Stratification of the top of the core due to chemical interactions with the mantle , journal Journal of Geophysical Research: Solid Earth volume 115 ( year 2010 )

    work page 2010

  5. [5]

    Brodholt , author J

    author J. Brodholt , author J. Badro , title Composition of the low seismic velocity e layer at the top of earth's core , journal Geophys. Res. Lett. volume 44 ( year 2017 ) pages 8303--8310 . https://agupubs.onlinelibrary.wiley.com/doi/10.1002/2017GL074261. :10.1002/2017GL074261

    work page doi:10.1002/2017gl074261 2017

  6. [6]

    Greenwood , author C

    author S. Greenwood , author C. J. Davies , author J. E. Mound , title On the evolution of thermally stratified layers at the top of earth's core , journal Physics of the Earth and Planetary Interiors volume 318 ( year 2021 ) pages 106763

    work page 2021

  7. [7]

    author C. J. Davies , author S. Greenwood , title Dynamics in earth's core arising from thermo-chemical interactions with the mantle , journal Core-Mantle Co-Evolution: An Interdisciplinary Approach ( year 2023 ) pages 219--258

    work page 2023

  8. [8]

    Alexandrakis , author D

    author C. Alexandrakis , author D. W. Eaton , title Precise seismic-wave velocity atop earth's core: No evidence for outer-core stratification , journal Physics of the Earth and Planetary Interiors volume 180 ( year 2010 ) pages 59--65

    work page 2010

  9. [9]

    author J. C. Irving , author S. Cottaar , author V. Leki \'c , title Seismically determined elastic parameters for earth’s outer core , journal Science advances volume 4 ( year 2018 ) pages eaar2538

    work page 2018

  10. [10]

    Kaneshima , title Array analyses of smks waves and the stratification of earth’s outermost core , journal Phys

    author S. Kaneshima , title Array analyses of smks waves and the stratification of earth’s outermost core , journal Phys. Earth Planet. Inter. volume 276 ( year 2018 ) pages 234--246

    work page 2018

  11. [11]

    Helffrich , author S

    author G. Helffrich , author S. Kaneshima , title Outer-core compositional stratification from observed core wave speed profiles , journal Nature volume 468 ( year 2010 ) pages 807--810

    work page 2010

  12. [12]

    author K. A. Whaler , title Does the whole of the earth's core convect? , journal Nature volume 287 ( year 1980 ) pages 528--530 . https://www.nature.com/doifinder/10.1038/287528a0. :10.1038/287528a0

    work page doi:10.1038/287528a0 1980

  13. [13]

    author K. A. Whaler , title Geomagnetic evidence for fluid upwelling at the core-mantle boundary , journal Geophysical Journal International volume 86 ( year 1986 ) pages 563--588 . https://academic.oup.com/gji/article-lookup/doi/10.1111/j.1365-246X.1986.tb03844.x. :10.1111/j.1365-246X.1986.tb03844.x

    work page doi:10.1111/j.1365-246x.1986.tb03844.x 1986

  14. [14]

    author H. Amit , title Can downwelling at the top of the Earth’s core be detected in the geomagnetic secular variation? , journal Physics of the Earth and Planetary Interiors volume 229 ( year 2014 ) pages 110--121 . https://linkinghub.elsevier.com/retrieve/pii/S0031920114000132. :10.1016/j.pepi.2014.01.012

    work page doi:10.1016/j.pepi.2014.01.012 2014

  15. [15]

    Lesur , author K

    author V. Lesur , author K. Whaler , author I. Wardinski , title Are geomagnetic data consistent with stably stratified flow at the core--mantle boundary? , journal Geophysical Journal International volume 201 ( year 2015 ) pages 929--946

    work page 2015

  16. [16]

    Huguet , author H

    author L. Huguet , author H. Amit , author T. Alboussi \`e re , title Geomagnetic dipole changes and upwelling/downwelling at the top of the earth's core , journal Frontiers in Earth Science volume 6 ( year 2018 ) pages 170

    work page 2018

  17. [17]

    Gubbins , title Geomagnetic constraints on stratification at the top of earth’s core , journal Earth, Planets and Space volume 59 ( year 2007 ) pages 661--664

    author D. Gubbins , title Geomagnetic constraints on stratification at the top of earth’s core , journal Earth, Planets and Space volume 59 ( year 2007 ) pages 661--664 . https://earth-planets-space.springeropen.com/articles/10.1186/BF03352728. :10.1186/BF03352728

    work page doi:10.1186/bf03352728 2007

  18. [18]

    Buffett , title Geomagnetic fluctuations reveal stable stratification at the top of the earth’s core , journal Nature volume 507 ( year 2014 ) pages 484--487

    author B. Buffett , title Geomagnetic fluctuations reveal stable stratification at the top of the earth’s core , journal Nature volume 507 ( year 2014 ) pages 484--487

    work page 2014

  19. [19]

    Landeau , author P

    author M. Landeau , author P. Olson , author R. Deguen , author B. H. Hirsh , title Core merging and stratification following giant impact , journal Nature Geoscience volume 9 ( year 2016 ) pages 786--789

    work page 2016

  20. [20]

    Gubbins , author C

    author D. Gubbins , author C. Thomson , author K. Whaler , title Stable regions in the earth's liquid core , journal Geophysical Journal International volume 68 ( year 1982 ) pages 241--251

    work page 1982

  21. [21]

    Labrosse , author J.-P

    author S. Labrosse , author J.-P. Poirier , author J.-L. Le Mou \"e l , title On cooling of the earth's core , journal Physics of the Earth and Planetary Interiors volume 99 ( year 1997 ) pages 1--17

    work page 1997

  22. [22]

    Pozzo , author C

    author M. Pozzo , author C. Davies , author D. Gubbins , author D. Alf \` e , title Thermal and electrical conductivity of iron at Earth's core conditions , journal Nature ( year 2012 ). www.nature.com/nature. :10.1038/nature11031

    work page doi:10.1038/nature11031 2012

  23. [23]

    author U. R. Christensen , title A deep dynamo generating mercury’s magnetic field , journal Nature volume 444 ( year 2006 ) pages 1056--1058

    work page 2006

  24. [24]

    author T. Nakagawa , title Effect of a stably stratified layer near the outer boundary in numerical simulations of a magnetohydrodynamic dynamo in a rotating spherical shell and its implications for earth’s core , journal Physics of the Earth and Planetary Interiors volume 187 ( year 2011 ) pages 342--352

    work page 2011

  25. [25]

    Christensen , title Geodynamo models with a stable layer and heterogeneous heat flow at the top of the core , journal Geophys

    author U. Christensen , title Geodynamo models with a stable layer and heterogeneous heat flow at the top of the core , journal Geophys. J. Int. volume 215 ( year 2018 ) pages 1338--1351 . https://academic.oup.com/gji/article/215/2/1338/5078871. :10.1093/gji/ggy352

    work page doi:10.1093/gji/ggy352 2018

  26. [26]

    Gastine , author J

    author T. Gastine , author J. Aubert , author A. Fournier , title Dynamo-based limit to the extent of a stable layer atop earth’s core , journal Geophys. J. Int. volume 222 ( year 2020 ) pages 1433--1448

    work page 2020

  27. [27]

    Aubert , title Rapid geomagnetic variations and stable stratification at the top of earth's core , journal Phys

    author J. Aubert , title Rapid geomagnetic variations and stable stratification at the top of earth's core , journal Phys. Earth Planet. Inter. volume 362 ( year 2025 ) pages 107335 . https://linkinghub.elsevier.com/retrieve/pii/S0031920125000299. :10.1016/j.pepi.2025.107335

    work page doi:10.1016/j.pepi.2025.107335 2025

  28. [28]

    Olson , author M

    author P. Olson , author M. Landeau , author E. Reynolds , title Dynamo tests for stratification below the core-mantle boundary , journal Phys. Earth Planet. Inter. volume 271 ( year 2017 ) pages 1--18 . https://linkinghub.elsevier.com/retrieve/pii/S0031920117301528

    work page 2017

  29. [29]

    Mound , author C

    author J. Mound , author C. Davies , author S. Rost , author J. Aurnou , title Regional stratification at the top of earth's core due to core–mantle boundary heat flux variations , journal Nat. Geosci. volume 12 ( year 2019 ) pages 575--580 . :10.1038/s41561-019-0381-z

    work page doi:10.1038/s41561-019-0381-z 2019

  30. [30]

    author J. E. Mound , author C. J. Davies , title Longitudinal structure of earth’s magnetic field controlled by lower mantle heat flow , journal Nat. Geosci. ( year 2023 ). :10.1038/s41561-023-01148-9

    work page doi:10.1038/s41561-023-01148-9 2023

  31. [31]

    author J. S. Turner , title Buoyancy effects in fluids , publisher Cambridge university press , year 1973

    work page 1973

  32. [32]

    Breuer , author A

    author M. Breuer , author A. Manglik , author J. Wicht , author T. Trümper , author H. Harder , author U. Hansen , title Thermochemically driven convection in a rotating spherical shell , journal Geophys. J. Int. volume 183 ( year 2010 ) pages 150--162 . :10.1111/j.1365-246X.2010.04722.x

    work page doi:10.1111/j.1365-246x.2010.04722.x 2010

  33. [33]

    Trümper , author M

    author T. Trümper , author M. Breuer , author U. Hansen , title Numerical study on double-diffusive convection in the earth's core , journal Phys. Earth Planet. Inter. volume 194-195 ( year 2012 ) pages 55--63 . :10.1016/j.pepi.2012.01.004

    work page doi:10.1016/j.pepi.2012.01.004 2012

  34. [34]

    Tassin , author T

    author T. Tassin , author T. Gastine , author A. Fournier , title Geomagnetic semblance and dipolar-multipolar transition in top-heavy double-diffusive geodynamo models , journal Geophys. J. Int. volume 226 ( year 2021 ) pages 1897--1919 . :10.1093/gji/ggab161

    work page doi:10.1093/gji/ggab161 2021

  35. [35]

    author J. E. Mound , author C. J. Davies , title Heat transfer in rapidly rotating convection with heterogeneous thermal boundary conditions , journal J. Fluid Mech. volume 828 ( year 2017 ) pages 601--629 . :10.1017/jfm.2017.539

    work page doi:10.1017/jfm.2017.539 2017

  36. [36]

    Bouffard , author G

    author M. Bouffard , author G. Choblet , author S. Labrosse , author J. Wicht , title Chemical convection and stratification in the earth's outer core , journal Frontiers in Earth Science volume 7 ( year 2019 ). https://www.frontiersin.org/article/10.3389/feart.2019.00099/full. :10.3389/feart.2019.00099

    work page doi:10.3389/feart.2019.00099/full 2019

  37. [37]

    Kutzner , author U

    author C. Kutzner , author U. R. Christensen , title From stable dipolar towards reversing numerical dynamos , journal Physics of the Earth and Planetary Interiors volume 131 ( year 2002 ) pages 29--45

    work page 2002

  38. [38]

    author T. G. Masters , author S. Johnson , author G. Laske , author H. Bolton , title A shear-velocity model of the mantle , journal Philos. T. Roy. Soc. London A volume 354 ( year 1996 ) pages 1385--1411

    work page 1996

  39. [39]

    Naskar , author C

    author S. Naskar , author C. Davies , author J. Mound , author A. Clarke , title Force balances in spherical shell rotating convection , journal J. Fluid Mech. volume 1009 ( year 2025 ) pages A70 . :10.1017/jfm.2025.259

    work page doi:10.1017/jfm.2025.259 2025

  40. [40]

    author M. A. Calkins , author R. J. Orvedahl , author N. A. Featherstone , title Large-scale balances and asymptotic scaling behaviour in spherical dynamos , journal Geophys. J. Int. volume 227 ( year 2021 ) pages 1228--1245 . :10.1093/gji/ggab274

    work page doi:10.1093/gji/ggab274 2021

  41. [41]

    author J. A. Nicoski , author A. R. O'Connor , author M. A. Calkins , title Asymptotic scaling relations for rotating spherical convection with strong zonal flows , journal J. Fluid Mech. volume 981 ( year 2024 ) pages A22 . :10.1017/jfm.2024.78

    work page doi:10.1017/jfm.2024.78 2024

  42. [42]

    author A. P. Willis , author B. Sreenivasan , author D. Gubbins , title Thermal core--mantle interaction: exploring regimes for ‘locked’ dynamo action , journal Phys. Earth Planet. Inter. volume 165 ( year 2007 ) pages 83--92

    work page 2007

  43. [43]

    author C. J. Davies , author D. Gubbins , author P. K. Jimack , title Scalability of pseudospectral methods for geodynamo simulations , journal Concurr. Comput. Pract. Exp. volume 23 ( year 2011 ) pages 38--56

    work page 2011

  44. [44]

    Matsui , author E

    author H. Matsui , author E. Heien , author J. Aubert , author J. M. Aurnou , author M. Avery , author B. Brown , author B. A. Buffett , author F. Busse , author U. R. Christensen , author C. J. Davies , et al., title Performance benchmarks for a next generation numerical dynamo model , journal Geochemistry, Geophys. Geosystems volume 17 ( year 2016 ) pag...

    work page 2016

  45. [45]

    author R. P. J. Kunnen , title The geostrophic regime of rapidly rotating turbulent convection , journal J. Turbul. volume 22 ( year 2021 ) pages 267--296

    work page 2021

  46. [46]

    author R. S. Long , author J. E. Mound , author C. J. Davies , author S. M. Tobias , title Scaling behaviour in spherical shell rotating convection with fixed-flux thermal boundary conditions , journal J. Fluid Mech. volume 889 ( year 2020 ) pages A7 . :10.1017/jfm.2020.67

    work page doi:10.1017/jfm.2020.67 2020

  47. [47]

    author J. E. Mound , author C. J. Davies , title Scaling laws for regional stratification at the top of earth's core , journal Geophys. Res. Lett. volume 47 ( year 2020 ). https://onlinelibrary.wiley.com/doi/10.1029/2020GL087715. :10.1029/2020GL087715

    work page doi:10.1029/2020gl087715 2020

  48. [48]

    Bouffard , author M

    author M. Bouffard , author M. Landeau , author A. Goument , title Convective Erosion of a Primordial Stratification Atop Earth's Core , journal Geophysical Research Letters volume 47 ( year 2020 ) pages e2020GL087109 . https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2020GL087109. :10.1029/2020GL087109

    work page doi:10.1029/2020gl087109 2020

  49. [49]

    Ma , author H

    author X. Ma , author H. Tkal c i \'c , title Seismic low-velocity equatorial torus in the earth’s outer core: Evidence from the late--coda correlation wavefield , journal Science Advances volume 10 ( year 2024 ) pages eadn5562

    work page 2024

  50. [50]

    author E. J. Kaplan , author N. Schaeffer , author J. Vidal , author P. Cardin , title Subcritical thermal convection of liquid metals in a rapidly rotating sphere , journal Phys. Rev. Lett. volume 119 ( year 2017 ) pages 094501 . https://link.aps.org/doi/10.1103/PhysRevLett.119.094501. :10.1103/PhysRevLett.119.094501

    work page doi:10.1103/physrevlett.119.094501 2017

  51. [51]

    Vidal , author N

    author J. Vidal , author N. Schaeffer , title Quasi-geostrophic modes in the earth's fluid core with an outer stably stratified layer , journal Geophysical Journal International volume 202 ( year 2015 ) pages 2182--2193 . https://doi.org/10.1093/gji/ggv282. :10.1093/gji/ggv282. http://arxiv.org/abs/https://academic.oup.com/gji/article-pdf/202/3/2182/16970...

    work page doi:10.1093/gji/ggv282 2015

  52. [52]

    author N. Schaeffer , title Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations , journal Geochemistry, Geophysics, Geosystems volume 14 ( year 2013 ) pages 751--758 . https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1002/ggge.20071. :https://doi.org/10.1002/ggge.20071

    work page doi:10.1002/ggge.20071 2013