the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Apr 2025
On stabilisation of compositional density jumps in compressible mantle convection simulations
Abstract. Large density jumps in numerical simulations of solid Earth dynamics can cause numerical "drunken sailor" oscillations. An implicit method has previously been shown to be very effective in stabilising the density jump that occurs at a free surface against such instabilities (Kaus et al., 2010; Duretz et al., 2011). Here the use of this to prevent oscillations of compositional layers deeper in the mantle is examined. If the stabilisation algorithm uses the total density field including the steady increase of density with depth due to adiabatic compression and jumps due to phase transitions then a severe artificial reduction of convective vigour occurs because the algorithm assumes that density is advected with the flow but these density gradients are not. This artificial vigour reduction increases with Rayleigh number but decreases with decreasing grid spacing. Thus, it is essential to use only composition-related density gradients in the stabilisation algorithm, and a simple method for isolating these is presented. Once this is done, the stabilisation method works effectively for internal compositional layers as well as a free surface.
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Status: closed
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RC1: 'Comment on egusphere-2025-1543', Mingming Li, 27 Jun 2025
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AC1: 'Reply on RC1', Paul Tackley, 27 Jun 2025
I'd like to thank the referee, Prof. Mingming Li, for his positive comments.
I will certainly make those small corrections and explain/derive equation (16) in a revised version.
Citation: https://doi.org/10.5194/egusphere-2025-1543-AC1
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AC1: 'Reply on RC1', Paul Tackley, 27 Jun 2025
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RC2: 'Comment on egusphere-2025-1543', Boris Kaus, 18 Aug 2025
This is a very interesting manuscript that points out that the stabilisation algorithm as originally proposed in Kaus et al. (2010) requires modifications before it can be applied to cases with adiabatic compression. It is well-written and logically argued, and the provided julia code works well.
I have only a few very minor suggestions (listed below), but would strongly suggest acceptance as soon once this is accounted for.
l. 25: Our original algorithm was in fact applied everywhere in the domain and not just at cells with a free surface. Obviously, since those cells have the largest density difference, most of the correction ended up being applied here.
l. 25: An additional paper that proposed a slightly different algorithm along with rigorous numerical stability analysis was by Rose et al. (2017), which is at least worth citing in this context. Also, fully implicit timestepping schemes can be used to circumvent the problem altogether as discussed by Popov and Sobolev (2008) and Kramer et al. (2012)
l. 147: Nusselt numbers are the identical => Nusselt numbers are identicla (remove “the”)
l. 147: the rms. velocity => the rms velocity (remove “.”)
l. 155: DGS => DJS
References
- Kramer, S.C., Wilson, C.R., Davies, D.R., 2012. An implicit free surface algorithm for geodynamical simulations. Physics of the Earth and Planetary Interiors 194–195, 25–37. https://doi.org/10.1016/j.pepi.2012.01.001
- Popov, A.A., Sobolev, S.V., 2008. SLIM3D: A tool for three-dimensional thermomechanical modeling of lithospheric deformation with elasto-visco-plastic rheology. Physics of the Earth and Planetary Interiors 171, 55–75. https://doi.org/10.1016/j.pepi.2008.03.007
- Rose, I., Buffett, B., Heister, T., 2017. Stability and accuracy of free surface time integration in viscous flows. Physics of the Earth and Planetary Interiors 262, 90–100. https://doi.org/10.1016/j.pepi.2016.11.007
Citation: https://doi.org/10.5194/egusphere-2025-1543-RC2 -
AC2: 'Reply on RC2', Paul Tackley, 18 Aug 2025
I thank Prof. Boris Kaus for his reviewer comments. I will certainly add the suggested discussion points / references to the revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-1543-AC2
Status: closed
-
RC1: 'Comment on egusphere-2025-1543', Mingming Li, 27 Jun 2025
This study solves an important numerical artefact when simulating mantle convection with significant density contrast between layers. The paper is well written and easy to follow. The results are supported by the experiments. I recommend ‘accept’ after a few very minor changes as detailed below.
Line 60, 118: ‘Where’ -> ‘where’
Line 108: \Theta is not defined in Eq. (13), or change \Theta to ‘the \theta in Equation (5)’
Equation 16: I do not quite understand this equation. Could you include some derivation for this equation?
Citation: https://doi.org/10.5194/egusphere-2025-1543-RC1 -
AC1: 'Reply on RC1', Paul Tackley, 27 Jun 2025
I'd like to thank the referee, Prof. Mingming Li, for his positive comments.
I will certainly make those small corrections and explain/derive equation (16) in a revised version.
Citation: https://doi.org/10.5194/egusphere-2025-1543-AC1
-
AC1: 'Reply on RC1', Paul Tackley, 27 Jun 2025
-
RC2: 'Comment on egusphere-2025-1543', Boris Kaus, 18 Aug 2025
This is a very interesting manuscript that points out that the stabilisation algorithm as originally proposed in Kaus et al. (2010) requires modifications before it can be applied to cases with adiabatic compression. It is well-written and logically argued, and the provided julia code works well.
I have only a few very minor suggestions (listed below), but would strongly suggest acceptance as soon once this is accounted for.
l. 25: Our original algorithm was in fact applied everywhere in the domain and not just at cells with a free surface. Obviously, since those cells have the largest density difference, most of the correction ended up being applied here.
l. 25: An additional paper that proposed a slightly different algorithm along with rigorous numerical stability analysis was by Rose et al. (2017), which is at least worth citing in this context. Also, fully implicit timestepping schemes can be used to circumvent the problem altogether as discussed by Popov and Sobolev (2008) and Kramer et al. (2012)
l. 147: Nusselt numbers are the identical => Nusselt numbers are identicla (remove “the”)
l. 147: the rms. velocity => the rms velocity (remove “.”)
l. 155: DGS => DJS
References
- Kramer, S.C., Wilson, C.R., Davies, D.R., 2012. An implicit free surface algorithm for geodynamical simulations. Physics of the Earth and Planetary Interiors 194–195, 25–37. https://doi.org/10.1016/j.pepi.2012.01.001
- Popov, A.A., Sobolev, S.V., 2008. SLIM3D: A tool for three-dimensional thermomechanical modeling of lithospheric deformation with elasto-visco-plastic rheology. Physics of the Earth and Planetary Interiors 171, 55–75. https://doi.org/10.1016/j.pepi.2008.03.007
- Rose, I., Buffett, B., Heister, T., 2017. Stability and accuracy of free surface time integration in viscous flows. Physics of the Earth and Planetary Interiors 262, 90–100. https://doi.org/10.1016/j.pepi.2016.11.007
Citation: https://doi.org/10.5194/egusphere-2025-1543-RC2 -
AC2: 'Reply on RC2', Paul Tackley, 18 Aug 2025
I thank Prof. Boris Kaus for his reviewer comments. I will certainly add the suggested discussion points / references to the revised manuscript.
Citation: https://doi.org/10.5194/egusphere-2025-1543-AC2
Model code and software
Software for manuscript "On stabilisation of compositional density jumps in compressible mantle convection simulations" (1.0) Paul Tackley and ETH Zurich https://doi.org/10.5281/zenodo.15115817
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This study solves an important numerical artefact when simulating mantle convection with significant density contrast between layers. The paper is well written and easy to follow. The results are supported by the experiments. I recommend ‘accept’ after a few very minor changes as detailed below.
Line 60, 118: ‘Where’ -> ‘where’
Line 108: \Theta is not defined in Eq. (13), or change \Theta to ‘the \theta in Equation (5)’
Equation 16: I do not quite understand this equation. Could you include some derivation for this equation?