## Abstract

In a uniform property spherical shell with the same inner to outer radius ratio, f, as the Earth's mantle, a bottom heating Rayleigh number, Ra, of 10^{7} and a nondimensional internal heating rate, H, of 23 (arguably Earth-like values) are insufficient to heat the mean temperature, θ, above the mean of the boundary value temperatures (nondimensional value 0.5). Thus, to attain spherical shell-type temperatures in a plane-layer geometry system, some degree of internal cooling is required. We present the findings from 68 calculations of three-dimensional plane-layer convection featuring a range of Rayleigh numbers and internal heating and cooling rates. Observed mean temperatures are fit with a power-law scaling and combined with the results from spherical shell geometry convection studies to obtain a single equation relating θ, Ra, H and f. For a given Rayleigh number, the derived expression can be used to calculate an appropriate heating or cooling rate for a plane-layer convection model in order to obtain the θ of a spherical system described by f. Encouragingly, we find that at a Rayleigh number consistent with estimates of the effective value of Ra for the Earth's mantle, geotherms are similar for an appropriately cooled plane-layer system and a spherical shell model featuring a value for H based on estimates of the present-day rate of mantle internal heating. In particular, our findings have important implications for plane-layer geometry numerical models of mantle convection featuring temperature-dependent parameters and laboratory tank models. For example, we conclude that disregarding internal heating is most appropriate for modelling terrestrial mantle convection in a plane-layer geometry. Moreover, because cooling a plane-layer model becomes increasingly relevant when emulating spherical shell convection at higher Rayleigh numbers, our study indicates important considerations for modelling super-Earth mantle dynamics in plane-layer convection studies.

Original language | English |
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Pages (from-to) | 73-84 |

Number of pages | 12 |

Journal | Physics of the Earth and Planetary Interiors |

Volume | 182 |

Issue number | 1-2 |

DOIs | |

State | Published - Sep 2010 |

### Bibliographical note

Funding Information:We thank the NSERC of Canada for funding the study of planetary mantle dynamics (#327084-2009).

## Keywords

- Geotherm
- Internal heating
- Mantle convection
- Rayleigh number
- Super-Earths
- Three-dimensional

## ASJC Scopus subject areas

- Astronomy and Astrophysics
- Geophysics
- Physics and Astronomy (miscellaneous)
- Space and Planetary Science