Comparison and verification of enthalpy schemes for polythermal glaciers and ice sheets with a one-dimensional model

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.