The representative elementary watershed (REW) approach proposed by Reggiani et al. was the first attempt to develop scale adaptable equations applicable directly at the macro scale. Tian et al. extended the initial definition of REW for simulating the energy related processes, and re-organized the deriving procedure of balance equations so that additional sub-regions and substances could be easily incorpo-rated. The resultant ordinary differential equation set can simulate various hydro-logical processes in a physically reasonable way. However, constitutive and geo-metric relationships have not been developed for Tian et al.'s equation set, which are necessary for the thermodynamic watershed hydrological model to apply in hydrological modeling practice. In this work, the constitutive equations for mass exchange terms and momentum exchange terms were developed as well as geo-metric relationships. The closed ordinary differential equation set with nine equa-tions was finally obtained.
Scale adaptable hydrological models have attracted more and more attentions in the hydrological modeling research community, and the constitutive relationship at the macro-scale is one of the most important issues, upon which there are not enough research activities yet. Taking the constitutive relationships of soil water movement--soil water retention curve (SWRC) as an example, this study extends the definition of SWRC at the micro-scale to that at the macro-scale, and aided by Monte Carlo method we demonstrate that soil property and the spatial distribution of soil moisture will affect the features of SWRC greatly. Furthermore, we assume that the spatial distribution of soil moisture is the result of self-organization of climate, soil, ground water and soil water movement under the specific boundary conditions, and we also carry out numerical experiments of soil water movement at the vertical direction in order to explore the relationship between SWRC at the macro-scale and the combinations of climate, soil, and groundwater. The results show that SWRCs at the macro-scale and micro-scale presents totally different features, e.g., the essential hysteresis phenomenon which is exaggerated with increasing aridity index and rising groundwater table. Soil property plays an important role in the shape of SWRC which will even lead to a rectangular shape under drier conditions, and power function form of SWRC widely adopted in hydrological model might be revised for most situations at the macro-scale.
