Ratio limits of water storage and outflow in a rainfall–runoff process

Abstract. Flash floods typically occur suddenly within hours of heavy rainfall. Accurate forecasting of flash floods in advance using the two-dimensional (2D) shallow water equations (SWEs) remains a challenge, due to the governing SWEs being difficult-to-solve partial differential equations (PDEs). Aiming at shortening the computational time and gaining more time for issuing early warnings of flash floods, constructing a new relationship between water storage and outflow in the rainfall–runoff process is attempted by assuming the catchment as a water storage system. Through numerical simulations of the diffusion wave (DW) approximation of SWEs, the water storage and discharge are found to be limited to envelope lines, and the discharge/water-depth process lines during water rising and falling showed a grid-shaped distribution. Furthermore, if a catchment is regarded as a semi-open water storage system, then there is a nonlinear relationship between the inside average water depth and the outlet water depth, namely, the water storage ratio curve, which resembles the shape of a plume. In the case of an open channel without considering spatial variability, the water storage ratio curve is limited to three values (i.e., the upper, the steady, and the lower limits), which are found to be independent of meteorological (rainfall intensity), vegetation (Manning's coefficient), and terrain (slope gradient) conditions. Meteorological, vegetation, and terrain conditions only affect the size of the plume without changing its shape. Rainfall, especially weak rain (i.e., when rainfall intensity is less than 5.0 mm h−1), significantly affects the fluctuations of the water storage ratio, which can be divided into three modes: Mode I (inverse S-shape type) during the rainfall beginning stage, Mode II (wave type) during the rainfall duration stage, and Mode III (checkmark type) during rainfall end stage. Results indicate that the determination of the nonlinear relationship of the water storage ratio curve under different geographical scenarios will provide new ideas for simulation and early warning of flash floods.