Porous flow of buoyant liquid through partially molten rock is regarded as the initial transport process leading to magma segregation in the mantle. Recent work has identified the importance of matrix deformation and compaction in this process. We present finite-difference calculations based on a generalized from of Darcy's law that includes matrix deformation, in two-dimensional Cartesian and cylindrical coordinates. We emphasize the existence of solitary wave solutions, called magmons. These waves are regions of locally high porosity that ascend through regions of low, uniform porosity. They differ from diapirs, where the liquid and matrix ascend together.

The one-dimensional waves that were previously reported are found to be unstable in two dimensions, breaking down to form two-dimensional waves of circular cross section. Both the development of the instability and the form of the two dimensional waves are supported by theoretical analysis. Numerical experiments in which two-dimensional waves collide show that the extent to which they are conserved after collision depends on the lateral offset of the waves. The conservation can be very good; in other cases, larger waves appear to grow at the expense of smaller waves. Although the quantification of the relevant permeability and rheological parameters remains uncertain, geophysically plausible estimates suggest wavelengths of kilometers and velocities of centimeters per year.

In our preliminary assessment of the relevance of these results to igneous processes, we find that magmons are unlikely to be important in regions of broad upwelling, such as beneath spreading centers. Here the transport of liquid adjusts to match the supply by melting, and compaction processes are not important. If liquid is supplied from below to a stable partially molten region of the asthenosphere, it is expected to ascend in magmons. This may occur beneath oceanic volcanic centers or in subcontinental mantle. It is possible that magma ascent in magmons leads to spatial and temporal episodicity of volcanic activity, if higher level processes do not obscure this influence. The waveform of a magmon ascends faster than the liquid within the magmon, so new liquid is taken in from above, while original liquid is lost from below. Consequently, magmons can mobilize small degrees of partial melt and deliver it rapidly to the surface.