  Accurate predictions of basal melt rates on ice shelves are necessary for
precise projections of the future behaviour of ice shelves. The computational
expense associated with completely resolving the cavity circulation using an
ocean model makes this approach unfeasible for multi-century simulations, and
parametrizations of melt rates are required. At present, some of the most
advanced melt rate parametrizations are based on a one-dimensional
approximation to the melt rate that emerges from the theory of subglacial
plumes under ice shelves with constant basal slopes and uniform ambient ocean
conditions; in this work, we present an asymptotic analysis of the
corresponding equations in which non-constant basal slopes and typical ambient
uniform conditions are imposed. This analysis exploits the small aspect ratio
of ice shelf bases, the relatively weak thermal driving and the relative
slenderness of the pycnocline in the ambient ocean. We construct an
approximation to the melt rate based on this analysis, which shows good
agreement with numerical solutions in a wide variety of cases suggesting a path
towards improved predictions of basal melt rates in ice-sheet models.
