The competition between percolation and circulation in a deformable porous medium

The equations governing the flow of a liquid in a viscously deformable porous medium are reexamined. Darcy's Law governs the “segregation,” the relative motion of the solid and liquid phases by percolation; the Stokes equation governs the “circulation,” the flow of both phases without relative motion. Terms describing the interaction between the segregation and circulation modify the familiar forms of these equations. In a previous contribution the approximation of omitting the circulation was adopted, and circular solitary wave solutions were found to be stable in two dimensions. Here we remove that approximation and present numerical experiments and analysis that indicate the continued existence of circular solitary waves. The consequences of these results in understanding the behavior of partially molten regions of the Earth's mantel is discussed. Regions of higher porosity may ascend either as diapirs (solid and liquid together) or as magmons (solitary waves, with segregation of solid and liquid). We argue that the characteristic length scale of magmons will always be superimposed on larger scale variations in porosity.