Thermal evolution of terrestrial planets with 2D and 3D geometries

<p>In mantle convection studies, two-dimensional geometry calculations are predominantly used, due to their reduced computational costs when compared to full 3-D spherical shell models.  Although various 3-D grid formulations [e.g. 1, 2] have been employed in studies using complex scenarios of thermal evolution [e.g., 3, 4], simulations with these geometries remain highly expensive in terms of computational power and thus 2-D geometries are still preferred in most of the exploratory studies involving broader ranges of parameters. However, these 2-D geometries still present drawbacks for modeling thermal convection. Although scaling and approximations can be applied to better match the average quantities obtained with 3D models [5], in particular, the convection pattern that in turn is critical to estimate melt production and distribution during the thermal evolution is difficult to reproduce with a 2D cylindrical geometry. In this scope, another 2D geometry called “spherical annulus” has been proposed by Hernlund and Tackley, 2008 [6]. Although steady state comparison between 2D cylindrical, spherical annulus and 3D geometry exist [6], so far no systematic study of the effects of these geometries in a thermal evolution scenario is available. </p><p>In this study we implemented a 2-D spherical annulus geometry in the mantle convection code GAIA [7] and used it along 2-D cylindrical and 3-D geometries to model the thermal evolution of 3 terrestrial bodies, respectively Mercury, the Moon and Mars. </p><p>First, we have performed steady state calculations in various geometries and compared the results obtained with GAIA with results from other mantle convection codes [6,8,9]. For this comparison we used several scenarios with increasing complexity in the Boussinesq approximation (BA).</p><p>In a second step we run thermal evolution simulations for Mars, Mercury, and the Moon using GAIA with 2D scaled cylinder, spherical annulus and 3D spherical shell geometries.In this case we considered the extended Boussinesq approximation (EBA), an Arrhenius law for the viscosity, a variable thermal conductivity between the crust and the mantle, while taking into account the heat source decay and the cooling of the core, as appropriate for modeling the thermal evolution. A detailed comparison between all geometries and planets will be presented focussing on the convection pattern and melt production. In particular, we aim to determine which 2D geometry reproduces most accurately the results obtained in a 3D spherical shell model. </p><p><strong>Aknowledgments</strong>: The authors gratefully acknowledges the financial support and endorsement from the DLR Management Board Young Research Group Leader Program and the Executive Board Member for Space Research and Technology.</p><p><strong>References</strong>: [1] Kageyama and Sato, G3, 2004; [2] Hüttig and Stemmer, G3, 2008;  [3] Crameri & Tackley, Progress Planet. Sci., 2016; [4] Plesa et al., GRL (2018); [5] Van Keken, PEPI, 2001; [6] Hernlund and Tackley, PEPI, 2008; [7] Hüttig et al, PEPI 2013; [8] Kronbichler et al., GJI, 2012; [9]  Noack et al., INFOCOMP 2015.</p>

Effect of grid resolution on tectonic regimes in global-scale convection models

<p>A key ingredient to reproduce plate-tectonics in numerical models is a viscoplastic rheology. Strongly temperature-dependent rheology generates a rigid lid at the surface, whereas plastic rheology allows for the formation of plate boundaries. The yield stress limiter  controls the strength of the lithosphere.</p><p>Depending on the value used for  different tectonics regimes can be observed: (i) dripping behaviour (low , (ii) plate-like behaviour (intermediate-low ), (iii) Episodic behaviour (intermediate-high ) and (iv) Stagnant lid behaviour (high ).</p><p>Each lid behaviour can be distinguished by comparing the evolution profile of several parameters: temperature, viscosity, surface Nusselt number and mobility (Tackley, 2000a.).</p><p>Despite the great importance of physical parameters, the outcome of geodynamical models is also affected by the grid resolution as it has been shown that the critical that separates each lid behavior is resolution dependent (Tosi et al., 2015).</p><p>Here we use the code StagYY (Tackley, 2008) in a 2D spherical annulus geometry (Hernlund & Tackley, 2008) to determine the resolution-dependent tectonic regime in a global-scale convection setting. We tested 12 grid resolutions (ranging from 128x32 to 1024x128 nodal points) and 9 different  (ranging from 10 to 90 MPa), keeping all the remaining physical parameters unchanged.</p><p>For these simplified models we assume isothermal free slip boundaries, constant radiogenic heating, no melting, endothermic (410) and exothermic (660) phase transitions. Each simulation was run for 15 Gyrs with a Rayleigh number of ≈8*10^7 to make sure that steady-state conditions were reached.</p><p>Our resolution tests show that the observed tectonic regime is affected by grid resolution as this parameter controls how well the lithosphere is resolved. Low radial resolutions favour weak lid regimes (dripping and plate-like) as the lithosphere is defined by few thick cells, that propagate basal stress to shallower depths. On the other hand low azimuthal resolutions favour strong lid regimes (episodic and stagnant) since plate boundaries remain unresolved. In conclusion, only at high grid resolutions (512x128 and higher) the numerical influence on the observed tectonic regime is low.</p>

The role of intrusive magmatism in shaping Venus’ present-day crust and its age distribution

<p>In its bulk properties, Venus appears similar to Earth, but both planets have developed substantially different geodynamic regimes. Earth has plate tectonics with a continuously renewed surface and its crustal distribution is very dichotomous in composition, thickness, and age. Venus, on the other hand, presently displays a period of a stagnant-lid regime, which may or may not was interrupted by catastrophic events of tectonic recycling during its history. Venus’ crustal thickness is not well constrained, but likely thicker than Earth’s oceanic crust; pronounced crustal dichotomy may be possible but evidence needs yet to be found. The age of the crust appears rather uniform, which traditionally has been taken as evidence that an episodic overturn must have taken place. However, recent arguments have challenged the episodic overturn hypothesis and favor a more continuous stagnant lid on Venus.</p><p> </p><p>To resolve the problem of Venus’ geodynamic regime understanding the generation of Venus’ crust in a dynamic context that also considers the underlying mantle is necessary. This can be achieved using numerical models of mantle convection tailored to Venus, which include the basic complexities of planetary mantle convection in terms of effective rheology, mineralogy and melting processes. Still, previous models have essentially failed to predict the thickness and age characteristics of Venus’ crust. One possible reason is that these models only considered extrusive volcanism, which renews the surface directly, while intrusive magmatism does not. Yet, intrusion seems the dominant mode of magmatism at least on Earth, so we investigate its influence in our model and evaluate whether this ingredient is key to predict Venus’ crustal characteristics.</p><p> </p><p>Using the code StagYY, we compute a suite of mantle convection models in 2D spherical annulus geometry that run through the entire solid-state history of Venus. We vary the partitioning of intrusive and extrusive volcanism from purely extrusive to dominantly intrusive and predict the present-day distributions of crustal thickness and surface age in the stagnant lid regime. With more intrusive magmatism, average crustal thickness is reduced by 20-25%, but mean crustal thickness still exceeds other independent estimates. The surface is on average much older, which is more consistent with mean age estimates from crater counting. However, lateral age variations also become stronger with dominantly intrusive volcanism, which indicates that volcanism keeps going on, but is more restricted spatially. Governing parameters like mantle reference viscosity and relative enrichment of heat-producing elements into the crust change the absolute values of mean crustal thickness and surface age, but do not improve surface age uniformity. This is somewhat at odds with Venus’ seemingly uniform surface age, so suitable conditions for this possibility are further evaluated in models featuring episodic overturn events.</p>

From a magma ocean to a solid mantle: implications for the thermo-chemical evolution of Mars

<p>Several studies suggest that Mars went through an episode of Magma Ocean (MO) early in its history. When the MO crystallises, solid mantle appears. The crystallisation of this MO starts at the Core-Mantle Boundary (CMB) and continues upwards to the surface of the planet. Assuming that this process occurs by fractional crystallisation, the solid cumulates that form are progressively enriched in incompatible elements, including iron, and an unstable density stratification is developed. This stratification is thought to have resulted in a planetary-scale mantle overturn after MO crystallisation, potentially explaining the early magnetic field, crustal dichotomy and chemical heterogeneities present on martian mantle.</p><p>However, previous studies on the thermo-chemical evolution of Mars consider only fractional crystallisation of the MO, and lack the possibility of re-melting/re-freezing of material at the mantle-MO interface, before the MO is fully crystallised.</p><p>In this study we investigate the effect of re-melting/re-freezing of material at the mantle-MO interface during MO crystallisation, on the dynamics and composition of the solid mantle. We use a numerical method with the convection code StagYY. The solid mantle is represented by a 2D spherical annulus geometry, and the MO by a 0D object at top of the mantle. The boundary condition applied to the solid domain allows the parameterisation of fractional crystallisation/re-melting of material at the mantle-MO interface. We model the growth of the solid mantle from the CMB up to the surface of the planet, and we account for core cooling and the presence of an atmosphere.</p><p>We show that by taking re-melting/re-freezing of material into account, the onset of convection can start earlier in Mars history. These results bring implications for the density stratification and overturn, and to the existence of isotopically distinct reservoirs on the mantle. Moreover, our results show that the mode of convection is preferentially degree-1, which can potentially explain the crustal dichotomy. </p>

Weakly nonlinear mode interactions in spherical Rayleigh–Bénard convection

In an annular spherical domain with separation $d$, the onset of convective motion occurs at a critical Rayleigh number $Ra=Ra_{c}$. Solving the axisymmetric linear stability problem shows that degenerate points $(d=d_{c},Ra_{c})$ exist where two modes simultaneously become unstable. Considering the weakly nonlinear evolution of these two modes, it is found that spatial resonances play a crucial role in determining the preferred convection pattern for neighbouring modes $(\ell ,\ell \pm 1)$ and non-neighbouring even modes $(\ell ,\ell \pm 2)$. Deriving coupled amplitude equations relevant to all degeneracies, we outline the possible solutions and the influence of changes in $d,Ra$ and Prandtl number $Pr$. Using direct numerical simulation (DNS) to verify all results, time periodic solutions are also outlined for small $Pr$. The $2:1$ periodic signature observed to be general for oscillations in a spherical annulus is explained using the structure of the equations. The relevance of all solutions presented is determined by computing their stability with respect to non-axisymmetric perturbations at large $Pr$.