  The enthalpy method for the thermodynamics of polythermal glaciers and ice
sheets is tested and verified by a one-dimensional problem (parallel-sided
slab). The enthalpy method alone does not include explicitly the transition
conditions at the cold-temperate transition surface (CTS) that separates the
upper cold from the lower temperate layer. However, these conditions are
important for correctly determining the position of the CTS. For the numerical
solution of the polythermal slab problem, we consider a two-layer
front-tracking scheme as well as three different one-layer schemes
(conventional one-layer scheme, one-layer melting CTS scheme, one-layer
freezing CTS scheme). Computed steady-state temperature and water-content
profiles are verified with exact solutions, and transient solutions computed by
the one-layer schemes are compared with those of the two-layer scheme,
considered to be a reliable reference. While the conventional one-layer scheme
(that does not include the transition conditions at the CTS) can produce
correct solutions for melting conditions at the CTS, it is more reliable to
enforce the transition conditions explicitly. For freezing conditions, it is
imperative to enforce them because the conventional one-layer scheme cannot
handle the associated discontinuities. The suggested numerical schemes are
suitable for implementation in three-dimensional glacier and ice-sheet models.
