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Modelling the porous flow of melt through a viscously deforming solid rock matrix is a useful tool for interpreting observations from the Earth's surface, and advances our understanding of the dynamics of the Earth's interior. However, the system of equations describing this process becomes mathematically degenerate in the limit of vanishing melt fraction. Numerical methods that do not consider this degeneracy or avoid it solely by regularizing specific material properties generally become computationally expensive as soon as the melt fraction approaches zero in some part of the domain. Here, we present a new formulation of the equations for coupled magma/mantle dynamics that addresses this problem, and allows it to accurately compute large-scale 3-D magma/mantle dynamics simulations with extensive regions of zero melt fraction. We achieve this by rescaling one of the solution variables, the compaction pressure, which ensures that for vanishing melt fraction, the equation causing the degeneracy becomes an identity and the other two equations revert to the Stokes system. This allows us to split the domain into two parts: in mesh cells where melt is present, we solve the coupled system of magma/mantle dynamics. In cells without melt, we solve the Stokes system as it is done for mantle convection without melt transport and constrain the remaining degrees of freedom. We have implemented this formulation in the open source geodynamic modelling code Aspect and illustrate the improved performance compared to the previous three-field formulation, showing numerically that the new formulation is robust in terms of problem size and only slightly sensitive to model parameters. Beyond that, we demonstrate the applicability to realistic problems by showing large-scale 2-D and 3-D models of mid-ocean ridges with complex rheology. Hence, we believe that our new formulation and its implementation in Aspect will prove a valuable tool for studying the interaction of melt segregating through and interacting with a solid host rock in the Earth and other planetary bodies using high-resolution, 3-D simulations. [ABSTRACT FROM AUTHOR]

Melt generation and migration are an important link between surface processes and the thermal and chemical evolution of the Earth's interior. However, their vastly different timescales make it difficult to study mantle convection and melt migration in a unified framework, especially for 3-D global models. And although experiments suggest an increase in melt volume of up to 20 per cent from the depth of melt generation to the surface, previous computations have neglected the individual compressibilities of the solid and the fluid phase. Here, we describe our extension of the finite element mantle convection code ASPECT that adds melt generation and migration. We use the original compressible formulation of the McKenzie equations, augmented by an equation for the conservation of energy. Applying adaptive mesh refinement to this type of problems is particularly advantageous, as the resolution can be increased in areas where melt is present and viscosity gradients are high, whereas a lower resolution is sufficient in regions without melt. Together with a high-performance, massively parallel implementation, this allows for high-resolution, 3-D, compressible, global mantle convection simulations coupled with melt migration. We evaluate the functionality and potential of this method using a series of benchmarks and model setups, compare results of the compressible and incompressible formulation, and show the effectiveness of adaptive mesh refinement when applied to melt migration. Our model of magma dynamics provides a framework for modelling processes on different scales and investigating links between processes occurring in the deep mantle and melt generation and migration. This approach could prove particularly useful applied to modelling the generation of komatiites or other melts originating in greater depths. The implementation is available in the Open Source ASPECT repository. [ABSTRACT FROM AUTHOR]