A spectral method to account for variable electrical conductivity in the dilute cores of the Gas Giants

Ever since the first satellite flybys of the Gas Giants, executed by the Pioneer and Voyager missions of the 1970s, the scope of the study of the interior structure and dynamics of Jupiter and Saturn has grown exponentially. Currently, with the wealth of data obtained from the Cassini-Huygens and Juno missions, we are able to construct more comprehensive models of these planets interior structures that would be able to explain and recreate some of their characteristic features, namely the general properties of their observed magnetic fields and zonal flows, and enable us to develop theories not only on planetary magnetism, but also on planetary formation, evolution, and our Solar System as a whole.Recent studies of the Jovian and Saturnian gravity fields suggest that both planets are likely to have dilute cores, characterised by an inhomogeneous heavy-element compositional gradient. This opens an opportunity to explore an alternative, full-sphere dynamo model, and compare it to previous standard models, which assume the presence of a distinct heavy-element inner core. Thus, the need arises for modified numerical interpretations, concerning boundary conditions and considerations of material properties, for the dynamo process in the presence of these dilute cores. To add to this, effects of a fully convective dilute core, double-diffusive convection, a more complex interior structure involving stably stratified layers, as well variable electrical conductivity are desired in order to consolidate the new model.To date, the majority of studies on electrical conductivity have assumed a near constant conductivity profile in the dynamo region, before having a steeply decaying conductivity in the outer envelope regions, in accordance with ab initio calculations. However, as part of the dilute core model, it is now all the more interesting to consider possible variations in the electrical conductivity profile, provided by the heavy-element gradient, in the dynamo region. This investigation may shine some light on any further potential complexities existing within the internal structures of Jupiter and Saturn, whether these add anything to our current models, and how these considerations ultimately affect the observed magnetic fields produced in simulations.To this end, an eigensolver based on the Tau Spectral Method, and using Jones-Worland polynomials as basis functions, is proposed to obtain the magnetic decay eigenmodes from the magnetic induction equation in the toroidal and poloidal decomposition, when there are no contributions from kinematic dynamo action (ie, the flow u=0). This eigensolver aims to show whether this method is indeed effective to apply to the case of a fully-spherical geometry with variable electrical conductivity, with the goal of comparing this methodology with the collocation method used in existing dynamo simulation codes, such as MagIC, and conclude its worth in the pursuit of further studies.