Topography and geoid induced by a convecting mantle beneath an elastic lithosphere

In the absence of seismological measurements, observations of the topography and gravity fields of solid planets are the primary constraints on their internal structure. To compute the synthetic geoid and topography induced by the dynamics of planetary interiors, we introduce a 3-D numerical tool describing mantle convection beneath an elastic lithosphere. Although the energy conservation is treated in the whole spherical domain, the deformation aspect is solved using a hybrid technique (finite volume method for the viscous flow, spectral method for elastic deformation). The mechanical coupling is achieved via the imposition of the traction at the surface of the viscous flow as a basal boundary condition for the elastic deformation. We present both response functions and full thermal convection cases computed with our new method for planetary bodies of varying dimensions: the filtering effect of the lithosphere on the dynamic topography and geoid is specific for each planetary body, justifying the importance of such a tool. Furthermore, since our approach specifically focuses on the mechanical coupling at the base of the lithosphere, it will permit future, more elaborate, rheological treatments. It also enables to discriminate between the radial and tangential components of the viscous traction. The latter is found to have a significant influence on the elastic deformation. The effect on geoid is prominent. More specifically, while a thin elastic lithosphere is usually considered to play little role on the dynamic topography and geoid of Venus, a ∼ 35 per cent reduction is obtained for geoid height in the numerical example we propose. On a planet with thicker elastic lithosphere such as Mars, the consequence of this filtering effect is to rule out the possibility of a dynamical support for the Tharsis Rise, even for the lowest admissible values of elastic thickness in this region. Key words: Lunar and planetary geodesy and gravity; Dynamics of lithosphere and mantle; Lithospheric flexure. ## 1. INTRODUCTION Thanks to the numerous spacecrafts that have orbited planetary bod ies in the last decades, the gravity field (derived from radio-tracking of the probes) and topography (measured by on-board altimeters, for example) are both known in great detail for all terrestrial planets but Mercury (see Wieczorek 2007, for a review of the knowledge on both data sets). These also start to be evaluated for icy moons although still mostly restricted to regional models for topography (e.g. on Europa, Nimmo et al. 2003) and to the smallest degrees of the gravitational field (cf. e.g. Iess et al. 2010, for Titan). Because the gravitational acceleration evaluated from the orbit is sensitive to the mass distribution within such bodies, its measurement may be used to invert parameters relevant for the structure of the interior. For this purpose, it is combined to topographic data given that sim plifying assumptions are postulated with regard to the deformation processes occurring within the planet. Wieczorek (2007) details how specific tools dedicated to the joint analysis of gravity and to pography fields can yield important insight into the knowledge of the interior. This insight can be crucial in the absence of seismic measurements since, in turn, it provides clues on the thermal state: Placing constraints on the internal structure of a planet, together with geological observations such as the evolution and distribution of volcanism helps to reconstruct the history of these solid bodies in terms of melting (including the presumably ancient formation of crust), deep internal processes (including early differentiation or the possible existence of a dynamo) and shallower geological processes. Because the spatial resolution associated with the gravity probe pertaining to the knowledge of planetary interiors decreases with depth, such studies are often dedicated to the shallower structure (e.g. the crust or the lithosphere): these include works on Mars C 2012 The Authors 55 (e.g. Neumann et al. 2004, on global variations of crustal thickness; and on the elastic lithosphere thickness beneath large volcanoes, Belleguic et al. 2005) and Venus (e.g. Simons et al. 1994, on the sta bility of crustal topography; and on the elastic lithosphere thickness at various locations, Barnett et al. 2002), but also on more distant bodies from very lacunar measurements, Europa (e.g. Nimmo et al. 2003), Thetys (Giese et al. 2007) or Rhea (Nimmo et al. 2010). Yet, the deeper interior might also contribute to the geoid and topography signatures at long wavelengths, through internal loading caused by large scale, deeply seated, density anomalies as well as through the viscous flow these may induce. Steinberger et al. (2010) propose that the largest wavelengths in the gravimetry and topography spectra (up to 30 for the Earth and Venus, up to 5 for Mars) are mostly attributable to sublithospheric structures. In this case of a deeper origin related to mantle dynamics and/or to large-scale compositional gradients, the rheological structure is a key ingredient. For example, Pauer et al. (2006) demonstrate in the case of Venus that two models explain equally well the data: (1) an isoviscous mantle beneath a strong and thick lithosphere of several hundreds of kilometres (as proposed by Kiefer et al. 1986), but also (2) a thinner lithosphere (∼ 100 km thick) and a viscosity increase with depth (between 10 and 100). Such an increase (either gradual or discontinuous) is also proposed for the Earth’s mantle on a comparable basis, to account for geoid highs above subduction zones (cf. e.g. the early works of Lago & Rabinowicz 1984; Hager et al. 1985). While the studies mentioned above are based on inversion proce dures invoking simple assumptions with regard to the deformation of the lithosphere and the underlying mantle, complementary works have computed synthetic observables from direct mantle flow cal culations (too complex to be yet introduced in an inversion frame work). A first group of studies considered simplified density and viscosity fields to understand the geoid above subduction zones for an isoviscous mantle (cf. Zhong & Gurnis 1992), for a vari able viscosity mantle (Moresi & Gurnis 1996) later introducing a 3-D spherical model (Zhang & Christensen 1993; Zhong & Davies 1999). A second group considered full thermal convection of a man tle whose viscosity is uniform or solely depth-dependent: these in clude works on Venus’ equatorial highlands (Kiefer & Hager 1991), on the Tharsis Rise (Harder & Christensen 1996), and, introducing depth-dependent viscosity in a spherical geometry (Harder 2000), as well as more generic studies such as the application to Earth’s hotspots and Venus’ coronae by Kiefer & Kellogg (1998). In their numerical investigation of long-wavelength geoid above subduction zones, King & Hager (1994) show however that it is important to include temperature-dependent viscosity in the convection experi ments. This is further demonstrated by Moresi & Parsons (1995) and Solomatov & Moresi (1996), the latter introducing a detailed description of the convective regimes in the context of temperature dependent viscosity. Both provide an application to the geoid and topography of Venus. Later investigations of the impact of variable viscosity convection on topography and geoid of planetary bodies include further studies of Mars (Redmond & King 2004; Kiefer & Li 2009), a regional study of the uplift in Beta Regio on Venus (Vezolainen et al. 2004) or a possible way of detecting convection at present within Mercury’s mantle (Redmond & King 2007). In these studies, the topographic signature of mantle flow is com puted from the vertical/radial normal stress induced by thermal convection (IVF, instantaneous viscous flow). However, the pres ence of a thick lithosphere can affect this prediction: in models of flexural loading (cf. Turcotte et al. 1981), the surface topogra phy results from the deformation of an elastic shell mimicking the rigidity of the lithosphere (see, Beuthe 2008, for a synthesis and the introduction of a model of lithospheric flexure for a spherical shell with a variable thickness). Zhong (2002) demonstrates that the IVF approach indeed overestimates the dynamical support of topogra phy for planets with a thick viscous lithosphere: A fully viscoelastic model is systematically compared to the IVF model showing that the steady-state viscous deformation might not be reached in geo logically relevant timescales. In the case of Mars, this result implies that the Tharsis Rise is unlikely to be dynamically supported (Zhong 2002; Zhong & Roberts 2003; Roberts & Zhong 2004). In these models, the loading associated with thermal convection is com puted indirectly: the distrib