Our knowledge of magma dynamics would be improved if geophysical data could be used to infer rheological constraints in melt-bearing zones. Geophysical images of the Earth's interior provide frozen snapshots of a dynamical system. However, knowledge of a rheological parameter such as viscosity would constrain the time-dependent dynamics of melt bearing zones. We propose a model that relates melt viscosity to electrical conductivity for naturally occurring melt compositions (including H2O) and temperature. Based on laboratory measurements of melt conductivity and viscosity, our model provides a rheological dimension to the interpretation of electromagnetic anomalies caused by melt and partially molten rocks (melt fraction ~ >0.7).

Nominally anhydrous minerals formed deep in the mantle and transported to the Earth’s surface contain tens to hundreds of ppm wt H₂O, providing evidence for the presence of dissolved water in the Earth’s interior. Even at these low concentrations, H₂O greatly affects the physico-chemical properties of mantle materials, governing planetary dynamics and evolution. The diffusion of hydrogen (H) controls the transport of H₂O in the Earth’s upper mantle, but is not fully understood for olivine ((Mg, Fe)₂SiO₄) the most abundant mineral in this region. Here we present new hydrogen self-diffusion coefficients in natural olivine single crystals that were determined at upper mantle conditions (2 GPa and 750–900 °C). Hydrogen self-diffusion is highly anisotropic, with values at 900 °C of 10^-10.9, 10^-12.8 and 10^-11.9 m²/s along [100], [010] and [001] directions, respectively. Combined with the Nernst-Einstein relation, these diffusion results constrain the contribution of H to the electrical conductivity of olivine to be σ_H = 10^2.12S/m·C_H20·exp^{−187kJ/mol/(RT)}. Comparisons between the model presented in this study and magnetotelluric measurements suggest that plausible H₂O concentrations in the upper mantle (≤250 ppm wt) can account for high electrical conductivity values (10^-2–10^-1 S/m) observed in the asthenosphere.