  In a number of geophysical or planetological settings (Earth's inner core, a
silicate mantle crystallizing from a magma ocean, or an ice shell surrounding a
deep water ocean) a convecting crystalline layer is in contact with a layer of
its melt. Allowing for melting/freezing at one or both of the boundaries of the
solid layer is likely to affect the pattern of convection in the layer. We
study here the onset of thermal convection in a viscous spherical shell with
dynamically induced melting/freezing at either or both of its boundaries. It is
shown that the behavior of each interface depends on the value of a dimensional
number P, which is the ratio of a melting/freezing timescale over a viscous
relaxation timescale. A small value of P corresponds to permeable boundary
conditions, while a large value of P corresponds to impermeable boundary
conditions. The linear stability analysis predicts a significant effect of
semi-permeable boundaries when the number P characterizing either of the
boundary is small enough: allowing for melting/freezing at either of the
boundary results in the emergence of larger scale convective modes. The effect
is particularly drastic when the outer boundary is permeable, since the degree
1 mode remains the most unstable even in the case of thin spherical shells. In
the case of a spherical shell with permeable inner and outer boundaries, the
most unstable mode consists in a global translation of the solid shell, with no
deformation. In the limit of a full sphere with permeable outer boundary, this
corresponds to the "convective translation" mode recently proposed for Earth's
inner core. As an example of possible application, we discuss the case of
thermal convection in Enceladus' ice shell assuming the presence of a global
subsurface ocean, and found that melting/freezing could have an important
effect on the pattern of convection in the ice shell.
