The Stability of A Coupled Energy-Water Balance System

The accurate initialization of land surface temperature, LST, (as the state variable of the surface energy balance equation) and soil moisture, SM, (as the state variable of the water balance equation) plays a vital role in the performance of land surface models (LSMs). The most common approach for initializing a LSM is to run it over a sufficiently long time in order to reach equilibrium values of SM and LST (model spin-up). Herein, a stability analysis is implemented on the temporal evolution of perturbed LST and SM to understand why these perturbations always dissipate in time and thus why a predefined equilibrium always exists for SM and LST states. The result of stability analysis shows that the dissipation of LST perturbations depends on SM anomalies, and that of soil moisture perturbations depends on LST anomalies; thus, the two systems are fully linked and each one affects the other. The dynamic equations governing LST and SM evolution are coupled and it is analytically shown why the coupled energy-water balance is strictly stable and always restores to an equilibrium state. Also, a time scale for restoring the coupled energy-water balance to a state of equilibrium in SM and LST is provided. It is found that the time required for SM and LST states to reach their equilibrium varies depending on the type of soil and climate.