  Liquid water oceans are thought to underlie the ice shells of Europa and
Enceladus. However, ocean properties can be challenging to measure due to the
overlying ice cover. Here, we explore how measurements of ice shell thickness,
which may be easier to obtain, could be used to infer information about the
subsurface ocean. In particular, we consider lateral gravity-driven flow of the
ice shells of icy satellites and relate this to ocean freeze and melt rates. We
employ a first-principles approach applicable to conductive ice shells. We
derive a scaling law under which ocean freeze and melt rates can be estimated
from thickness measurements of a shell with a vertically-varying
temperature-dependent viscosity. Under a steady-state assumption, ocean freeze
and melt rates can be inferred from measurements of ice thickness; however,
these rates depend on the basal viscosity, a key unknown. Depending on a
characteristic thickness scale and basal viscosity, the characteristic freeze
and melt rates range from about O(10$^{-1}$) to O(10$^{-5}$) mm/year. We
validate our scaling in an Earth environment with ice-penetrating radar
measurements of ice thickness and modelled snow accumulation for Roosevelt
Island, Antarctica. Our model, coupled with the forthcoming observations of
shell thickness from upcoming missions, may help bound the magnitudes of
estimated ocean freeze and melt rates on icy satellites and shed light on
potential ocean stratifications.
