Improving the Numerical Solution of the Energy Equation in Land Models

Abstract The purpose of this study is to improve simulations of the energy equation in land models. Our objectives are to (a) highlight the shortcomings of current approaches for solving the energy equation in land models; (b) present improved solutions; and (c) evaluate tradeoffs between numerical errors and strict energy conservation in the energy equation. We use the Structure for Unifying Multiple Modeling Alternatives land model to evaluate five approaches to solve the energy equation, including approaches that do not use time integration methods with rigorous error control (as is common in land models) as well as approaches that do. Simulations over North America show that numerical solutions of the energy equation in the form most commonly used in land models produce both large violations in energy conservation (especially in cold regions) as well as large numerical errors in soil temperature and soil water content. Alternative approaches for solving the energy equation demonstrate the tradeoff between strictly conserving energy and reducing overall numerical errors. The mixed form of the energy equation can be solved to conserve energy to within machine precision. The direct solution of the energy equation (i.e., using enthalpy as a primary variable) yields the smallest numerical errors, and except for the mixed form of the energy equation, the direct solution has the smallest errors in energy conservation. With the energy equation at the core of land models, improving its numerical solution is essential to accelerating future model development initiatives for other terrestrial system processes.