Deniz Soyuer
Determining the depth of atmospheric winds in the outer planets of the Solar system is a key topic in planetary science. We provide constraints on these depths in Uranus and Neptune via the total induced Ohmic dissipation, due to the interaction of the zonal flows and the planetary magnetic fields. An upper bound can be placed on the induced dissipation via energy and entropy flux throughout the interior. The induced Ohmic dissipation is directly linked to the electrical conductivity profile of the materials involved in the flow. We present a method for calculating electrical conductivity profiles of ionically conducting hydrogen–helium–water mixtures under planetary conditions, using results from ab initio simulations. We apply this prescription on several ice giant interior structure models available in the literature, where all the heavy elements are represented by water. According to the energy (entropy) flux budget, the maximum penetration depth for Uranus lies above 0.93 R (0.90 R) and for Neptune above 0.95 R (0.92 R). These results for the penetration depths are upper bounds and are consistent with previous estimates based on the contribution of the zonal winds to the gravity field. As expected, interior structure models with higher water abundance in the outer regions also have a higher electrical conductivity and therefore reach the Ohmic limit at shallower regions. Thus, our study shows that the likelihood of deep-seated winds on Uranus and Neptune drops significantly with the presence of water in the outer layers.
This question has been asked before for the other giant planets in our neighbourhood; Jupiter and Saturn. It has been shown that zonal winds on the gas giants are indeed a feature of their shallow layers. Two independent constraints were placed on the wind depths; one coming from the density anomalies due to the zonal flows and their projection onto the external gravity field measurements (carried out by Juno and Cassini spacecraft). The other coming from the heat generation due to the induced currents from the interaction between the planetary magnetic field and hypothesised deep-seated flows. Here, we use the latter constraint to estimate the maximum penetration depth of the zonal flows on Uranus and Neptune.
Since both the magnetic field strength and the electrical conductivity increase with depth, the induced currents and thus the dissipation is expected to increase dramatically as well (potentially outshining the planet). Therefore, one can calculate for a given interior structure model, bulk composition and wind behaviour, whether the total induced Ohmic dissipation is allowed in the heat flux budget (or the entropy flux budget) of the system.
Unlike that of the gas giants, the compositions of the ice giants are poorly understood. Jupiter and Saturn are mainly composed of hydrogen, which becomes semi-conducting already in the shallow layers of the planets. The electrical conductivity inside the gas giants therefore increases almost exponentially with depth, constraining the maximum penetration depth of the winds around 0.96 R in Jupiter and 0.86 R in Saturn.
The ice giants are hypothesised to have so-called ‘liquid ices’, which are neither liquids nor ices, but rather a complex mixture of superionic fluids, hydrogen and helium. We present a method for calculating electrical conductivity profiles of hydrogen—helium—water mixtures under planetary conditions. Unlike the ‘electronic’ conductivity of semi-conducting hydrogen seen in Jupiter and Saturn, the main channel of electrical conduction in this mixture is ‘ionic’. We construct our prescription starting from the fluctuation—dissipation theorem, where we take into account the various species that are present in the mixture (and their abundance depending on the interior structure of the planets). By modelling the diffusivity of each species and the effective charge of hydrogen ions, we calculate radial electrical conductivity profiles for various ice giant interior structure models in the literature.
The radial electrical conductivity estimates allow us to calculate the total induced Ohmic dissipation with depth. We find that according to the energy (entropy) flux budget, the maximum penetration depth for Uranus lies above 0.93 R (0.90 R) and for Neptune above 0.95 R (0.92 R).