Thermo-viscous fingering of flow in a thin gap: A model of magma flow in dikes and fissures

Flow of a fluid with a strongly temperature-dependent viscosity in a finite-length slot is analysed as a model of magma flow in dikes. The slot walls are held at a fixed temperature, thus cooling and increasing the viscosity of the fluid as it moves along the gap. Poiseuille flow and temperature advection, averaged across the slot, are used to study the stability of this basic one-dimensional flow to lateral perturbations. A linear stability analysis shows that for sufficiently strong cooling and viscosity increase with decreasing temperature, the flow is unstable to a fingering instability. Warm fluid is focused into relatively fast flowing zones and suffers only modest cooling, while cold, slow flowing regions experience more cooling and an increase in viscosity, which acts to locally clog the slot. The necessary condition for instability is the presence of multiple solutions for velocity (fast, intermediate and slow branches) in the basic one-dimensional flow. The intermediate branch, where the thermal adjustment lengthscale is comparable to the slot length, is unstable and the analysis indicates that the instability continues onto the slow branch. The parametric regions of instability and the growth rates are dependent on the choice of boundary conditions at the slot entrance (i.e. the magma source): either uniform flux, or uniform pressure. The latter case is the more geophysically realistic and has the larger unstable region and growth rates. Numerical solutions of the nonlinear equations show that at finite-amplitude the hot, low-viscosity, fast-flowing fingers continue to speed up, while the slow, cold regions continue to cool and slow down. At the slot exit fluid issues from the gap in isolated hot, low-viscosity spouts separated by zones of cold, nearly still fluid. Application of the model to geophysical settings indicates that the instability is expected for realistic parameter values. The model may help explain the observed focusing of fissure eruptions.