  Motivated by the mushy zones of sea ice, volcanoes, and icy moons of the
outer solar system, we perform a theoretical and numerical study of
boundary-layer convection along a vertical heated wall in a bounded ideal mushy
region. The mush is comprised of a porous and reactive binary alloy with a
mixture of saline liquid in a solid matrix, and is studied in the near-eutectic
approximation. Here we demonstrate the existence of four regions and study
their behavior asymptotically. Starting from the bottom of the wall, the four
regions are (i) an isotropic corner region; (ii) a buoyancy dominated vertical
boundary layer; (iii) an isotropic connection region; and (iv) a horizontal
boundary layer at the top boundary with strong gradients of pressure and
buoyancy. Scalings from numerical simulations are consistent with the
theoretical predictions. Close to the heated wall, the convection in the mushy
layer is similar to a rising buoyant plume abruptly stopped at the top, leading
to increased pressure and temperature in the upper region, whose impact is
discussed as an efficient melting mechanism.
