Unified theory of electrical conduction in firn and ice: Site percolation and conduction in snow and firn

Abstract

The conduction process in firn can be modeled by the theory of percolation and conduction in disordered systems. Effectively, firn is an infinite cluster of ice crystals with a statistically defined geometry. The properties of such clusters can be studied by random resistor network lattices, where the connectivity of the continuous conducting material can be controlled stochastically. These mixtures of conducting and insulating materials show a percolation threshold at which the conduction abruptly ceases. The structure of cold polar solid ice shows one ice crystal packed by 14 to 16 neighbors. As the density decreases, the number of nearest neighbors diminishes, which reduces the number of clusters that could contribute to the conduction process. From the volume fraction of ice at pore closeoff and the symmetry of the firn we obtain a critical volume fraction ≃0.08 (density equal to 0.07 Mg m−3) at which the conductivity should vanish. This agrees well with estimates of the critical volume fraction from percolation theory for a Bethe lattice with coordination number z = 14–16 and for a seven‐dimensional (7‐D) to eight‐dimensional (8‐D) hypercubic lattice (for which also z = 14–16). Conductivity in random networks obeys the power law σ(χ) ∝ (χ‐χc)t, where χ and χc are the volume fraction and critical fraction, respectively. The exponent t depends on the geometry and dimensionality of the lattice. We find that for firn, t = 2.7, in good agreement with both exponents calculated numerically for 7‐D to 8‐D hypercubic lattices and exponents obtained by extrapolation of theoretical values of t. These experimental observations and analyses explain why Looyenga's empirical mixture equation approximates the transport properties of firn. Furthermore, the analyses indicate that for the current to be transported from one crystal to up to 14 to 16 neighbors the ionic impurities must either be located in the ice lattice by substitution (i.e., with bulk conduction taking place) or form a coating of impurities that surround the crystal like a shell. These results do not support the concept of conduction along three‐grain boundaries, which implies z ≃ 4. In that case, according to site percolation theory, the conductivity would have to vanish at densities less than ∼0.4 Mg m−3, and the conductivity exponent t would be less than 1.8. These values are incompatible with measured conductivities in firn. Furthermore, it is physically impossible to form three‐grain boundaries at low densities.