  We study a semi-discrete model for the two-dimensional incompressible porous
medium (IPM) equation describing gravitational fingering phenomenon. The model
consists of a system of advection-reaction-diffusion equations on
concentration, velocity and pressure, describing motion of miscible liquids
under the Darcy's law in two vertical tubes (real lines) and interflow between
them. Our analysis reveals the structure of gravitational fingers in this
simple setting - the mixing zone consists of space-time regions of constant
intermediate concentrations and the profile of propagation is characterized by
two consecutive traveling waves which we call a terrace. We prove the existence
of such a propagating terrace for the parameters corresponding to small
distances between the tubes. This solution shows the possible mechanism of
slowing down the fingers' growth due to convection in the transversal
direction. The main tool in the proof is a reduction to pressure-free
transverse flow equilibrium (TFE) model using geometrical singular perturbation
theory and the persistence of stable and unstable manifolds under small
perturbations.
