The ground-state conductivity profile of the core-mantle system of the Earth

The ground state of the core-mantle conductivity system is defined as a step function transition from the constant non-zero conductivity of the core to zero conductivity in the mantle. This ground state is reached by letting the thickness of a transition region Δ → 0, where Δ appears as a parameter in the model conductivity of the system. By a transformation of the equation governing the behavior of the electrostatic potential φ, it is shown that the function V(x) = (1/σ^1/2)(d²σ^1/2/dx²) acts as a potential barrier in the quantum mechanical sense, and that for certain conductivity profiles σ(x), where x is the usual Cartesian coordinate, the electrostatic potential is screened out in regions where σ(x) → 0 as Δ → 0. Consequently, E = ?φ also vanishes in these regions. The results generalize to the time-dependent case. Conditions that the conductivity function σ(x) must satisfy to qualify as the ground-state conductivity are defined and an example is provided.