Improving the numerical solution of the energy equation in land models

At its core, a hydrological model is comprised of the conservation of energy and mass for a myriad of modeled processes across a suite of spatio-temporal scales. Simulations over North America with the SUMMA hydrological model show that the form of the energy equation most commonly used in land models produces both large violations in energy conservation (especially in cold regions) as well as larger numerical errors in soil temperature and soil water content than is possible with more robust solvers. These numerical issues sabotage the success of efforts to improve process-representation. We present improved energy-conserving solutions for land models, testing five approaches over North America with the SUMMA model and evaluating tradeoffs between strict energy conservation and numerical errors in the energy equation. We include approaches that do not use time integration methods with rigorous error control (as is common in hydrological models) as well as approaches that do. The mixed form of the energy equation is discretized to conserve energy to within machine precision. Alternatively, the direct solution of the energy equation (i.e., using enthalpy as a primary variable) yields the smallest numerical errors because it allows error control to be placed on the inherent state variable. In the spirit of advancing process-representation, we illustrate the importance of accurate energy balance solutions for simulations of partially frozen soils, permafrost, and glaciers. In one prominent example, we demonstrate that debris-covered glaciers have substantially dissimilar runoff contributions when evolved using different solutions to the energy equation. The capability to accurately simulate the energy balance of terrestrial systems is essential to improve the theoretical underpinnings of process-based hydrologic models.