Estimates for the Effective Electrical Conductivity of the Core in the Interior of Jupiter and Saturn
In the deep interior of the giant planets Jupiter and Saturn, ordinary hydrogen and helium are transformed into a conducting metallic liquid at extremely high pressure. It is likely that the giant planets' observed magnetic field is constantly generated in the metallic fluid core by magnetohydrodynamic processes, converting mechanic energy in the form of convection into magnetic energy. The maximum strength of their magnetic fields is likely to be limited by magnetic field instabilities which convert the magnetic energy back into convection. The parameter which governs the occurrence of magnetic instabilities is the Elsasser number, λ = B 2Σ/2Ωϱ, where B is the field strength, Σ is the electrical conductivity, Ω is the rotation rate and ϱ is the density. Since magnetic instability will be very active when λ exceeds a critical value λ c ∼ 10 (the precise value depending on the magnetic field distribution), this imposes an upper bound on the effective electrical conductivity of the metallic fluid which comprises the bulk of Jupiter's interior and much of Saturn's. Stability calculations including both toroidal (model) and poloidal (observed) components of the magnetic field in a rapidly rotating spherical shell, have been performed. The most stable configuration of the field is when the poloidal component of field is strong and the toroidal field is weak; in this case we obtain an upper bound for electrical conductivity of Σ ∼ 3 × 106 S/m; while the most unstable configuration of the field is when the toroidal and poloidal fields are comparable, giving rise to Σ m ∼ 3 × 105 S/m. The implications of the results for general dynamo theory are also discussed.