Calculating actual crop evapotranspiration under soil water stress conditions with appropriate numerical methods and time step Songhao Shang* State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China Abstract: Calculation of actual crop evapotranspiration under soil water stress conditions is crucial for hydrological modeling and irrigation water management. Results of actual evapotranspiration depend on the estimation of water stress coefficient from soil water storage in the root zone, which varies with numerical methods and time step used. During soil water depletion periods without irrigation or precipitation, the actual crop evapotranspiration can be calculated by an analytical method and various numerical methods. We compared the results from several commonly used numerical methods, including the explicit, implicit and modified Euler methods, the midpoint method, and the Heun’s third-order method, with results of the analytical method as the bench mark. Results indicate that relative errors of actual crop evapotranspiration calculated with numerical methods in one time step are independent of the initial soil water storage in the range of soil water stress. Absolute values of relative error decrease with the order of numerical methods. They also decrease with the number of time step, which can ensure the numerical stability of successive simulation of soil water balance. Considering the calculation complexity and calculation errors caused by numerical approximation for different time step and maximum crop evapotranspiration, the explicit Euler method is recommended for the time step of 1 day (d) or 2 d for maximum crop evapotranspiration less than 5 mm/d, the midpoint method or the modified Euler method for the time step of up to one week or 10 d for maximum crop evapotranspiration less than 5 mm/d, and the Heun’s third- order method for the time step of up to 15 d. Copyright © 2011 John Wiley & Sons, Ltd. KEY WORDS evapotranspiration; water stress; time step; explicit Euler method; midpoint method; Heun’s third-order method Received 30 April 2011; Accepted 1 November 2011 INTRODUCTION Water stress is common for crops in many places of the world where there is insufficient precipitation and irrigation. It has great impact on crop water use, and consequently on crop growth and yield. The calculation of actual crop evapotranspiration under water stress conditions is crucial for modeling soil water balance and its impact on crops and other terrestrial ecosystems (Daly and Porporato, 2005), and for irrigation water management (Muralidharan and Knapp, 2009). Crop water stress is often expressed as the ratio of actual to maximum crop evapotranspiration, which is called water stress coefficient (WSC). WSC is mainly influenced by soil water regime and soil properties, and is also related with maximum crop evapotranspiration. It is usually assumed to be 0 and 1 when soil water content in the root zone is less than the water content at wilting point (W p ) and greater than a critical water content for soil water stress (W j ), respectively (e.g. Shuttleworth, 1993; Allen et al., 1998). When soil water content increases from W p to W j , WSC increases from 0 to 1 and can be calculated with various types of empirical formulae. For some empirical formulae, WSC was calculated from available soil moisture using the logarithmic function (Jensen et al., 1970), the power function (Kang et al., 1994), or the exponential function (Poulovassilis et al., 2001), or from soil matric potential using a sigmoid function (Muralidharan and Knapp, 2009). The influence of maximum crop evapotranspiration was also considered in some formulae (e.g. Kristensen and Jensen, 1975; Allen et al., 1998). After choosing appropriate empirical formula to calculate WSC, crop evapotranspiration and soil water balance can be simulated in successive time steps. Since soil water content varies within a time step, a problem related with WSC calculation is how to choose an appropriate value of soil water content within each time step. If the time step is shorter, such as a day, the variation of soil water content within the time step is small, and WSC can be calculated with the initial soil water content (Shang and Mao, 2006). However, the time step is usually a week, 10 days, or even a growing stage in irrigation water management (Rao et al., 1988). In cases of longer time steps, the variation of soil water content within a time step is not negligible, and WSC should be calculated with caution to ensure the precision and numerical stability of evapotranspiration calculation. In this paper, actual crop evapotranspiration under soil water stress conditions was calculated with various *Correspondence to: Songhao Shang, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China. E-mail: shangsh@tsinghua.edu.cn HYDROLOGICAL PROCESSES Hydrol. Process. (2011) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/hyp.8405 Copyright © 2011 John Wiley & Sons, Ltd.