The 3D Development of Magmatic Plumbing Systems by the Melt-Back of Dike Walls

Dikes are magma filled tension cracks that can propagate out of magmatic centers on distance scales of meters to thousands of kilometers. Observations of basaltic eruptions in Iceland and Hawaii suggest that sustained lateral flow of magma over periods of days to months can produce widening of the deeper part of the dike due to melt-back of the dike walls. Numerical simulation of this process requires treatment of heat transfer in three dimensions. We describe a new 3D treatment of dike related advection and diffusion of heat using a simple finite difference approach. This effort builds on 2D models of dike freezing and melt-back and includes temperature-dependent viscosity of the magma. A key assumption is that the flow of magma in the dike can be described as viscous channel flow and that the dike center is a plane. Advection is the dominant mechanism of heat transfer in that plane and diffusion is the only significant mechanism of heat transfer orthogonal to the dike. A previous model of dike wall melt-back made the simplifying assumption that flow streamlines are parallel to the dike walls. This is inconsistent with conservation of mass and energy and the new treatment shows that this assumption significantly affects the time to freeze a dike. The 3D version of the new model shows that nearly horizontal magmatic "pipes" can develop in the deeper part of a dike on a timescales of days to weeks for mechanically reasonable dike opening geometries and magma fluxes.