Onset of convection in a basally heated spherical shell, application to planets

Convective instabilities related to the early dynamics of planetary mantles just after core formation play an important role in the subsequent evolution. Although these early stages of planetary dynamics are likely to imply more complex phenomena such as global melting and fractional solidification, and although density variations of compositional origin are likely to play an important role, little is known about the onset of solid-state convection in a fluid with temperature-dependent viscosity heated from below. Here, we investigate onset times of convection in order to obtain scaling relationships for the influences of Rayleigh number, viscosity parameter describing the dependency on the temperature and geometry of spherical shell (measured by f  , ratio between the inner and outer radii). We performed three-dimensional numerical experiments and we concentrate on the dynamical regime described by global viscosity contrasts smaller than 104104. Onset times and wavelengths of the first instabilities using both dynamical (free-slip) and kinematical (no-slip) boundary conditions are investigated. For both boundary conditions, the scaling may be written in the form t′∝(Ra∗)at′∝(Ra∗)a, where a   is approximately −2/3−2/3 and Ra∗=Ra(μ(θ∗))Ra∗=Ra(μ(θ∗)) is a Rayleigh number specifically associated with a relevant temperature (viscosity) value (θ∗≈0.25θ∗≈0.25). In addition, the dimensionless onset times (using the shell thickness as a characteristic length scale) are almost independent on the geometry of the shell for large range of the geometrical factor (f≥0.2f≥0.2). In order to better understand these processes, 3D results are compared with two simple methods: the linear stability (LS) analysis and the growth of Rayleigh–Taylor (R–T) instabilities. The LS analysis values of the onset times are much smaller due to the “frozen time” approach (i.e. the conductive propagation of the hot front is not taken into account). The dependency of the onset time on the Rayleigh number is overestimated, especially for the free-slip conditions, where the “frozen time” effect is even more significant. For the R–T instability analysis, although the onset times are also underestimated, the agreement with 3D simulations is good in terms of efficient scaling relationships. When applied to the dimensions and plausible initial state of terrestrial planets (Mars, the Earth and Venus), the scaling relationships provide an idealized framework to investigate early dynamics. Due to uncertainties associated with the “initial” temperature field and viscosity parameters, the computed onset times vary by several orders of magnitude (between 0.1 Myr and 500 Myr). These are likely to be smaller than the ones obtained for the onset of convection at the base of the lithosphere. For the investigated range of parameters, the minimal preferred degree for the onset instabilities is estimated to be approximately 10 so that, other ingredients or a different dynamical regime, have to be considered to promote the very low degree convective instabilities suggested for the early evolution of Mars.