New analytical derivation of the mean annual water‐energy balance equation

Abstract

[1] The coupled water-energy balance on long-term time and catchment scales can be expressed as a set of partial differential equations, and these are proven to have a general solution as E/P = F(E 0 /P, c), where c is a parameter. The state-space of (P, E 0 , E) is a set of curved faces in P À E 0 À E three-dimensional space, whose projection into E/P À E 0 /P two-dimensional space is a Budyko-type curve. The analytical solution to the partial differential equations has been obtained as E = E 0 P/(P n + E 0 n ) 1/n (parameter n representing catchment characteristics) using dimensional analysis and mathematic reasoning, which is different from that found in a previous study. This analytical solution is a useful theoretical tool to evaluate the effect of climate and land use changes on the hydrologic cycle. Mathematical comparisons between the two analytical equations showed that they were approximately equivalent, and their parameters had a perfectly significant linear correlation relationship, while the small difference may be a result of the assumption about derivatives in the previous study.

Determining the hydrological responses to climate variability and land use/cover change in the Loess Plateau with the Budyko framework

“…These two equations were shown to be virtually identical, and the Choudhury equation can be derived using Fu's approach (Yang et al, 2008). In this study, the Fu equation was used (Fu, 1981;Zhang et al, 2004):…”

Comparing two tools for ecosystem service assessments regarding water resources decisions

“…The InVEST water model uses a Budyko curve approach [Budkyo, 1974;Zhang et al, 2004] with a Fu-type equation [Yang et al, 2008] to calculate the average annual water yield for each subbasin based upon the vegetation and aridity of the watershed, and assuming no change in storage. The InVEST water model includes two main components: (1) a dryness index, which includes reference evapotranspiration, annualized crop coefficient, and precipitation; and (2) a storage component which includes soil depth, plant root depth, soil water storage, and a seasonality term.…”

Analysis of catchment evapotranspiration at different scales using bottom-up and top-down approaches

“…Budyko (1974) assumed that actual evapotranspiration is controlled by both water and energy availabilities and considered that the ratio of annual evapotranspiration and annual precipitation is the function of the ratio of annual precipitation and annual net radiation. Based on the Budyko hypothesis, and based on dimensional analysis and mathematical reasoning, Yang et al (2008a) considered the important effect of the vegetation and derived an analytical equation of the coupled water-energy balance at an annual time scale, expressed as:…”

“…Budyko (1974) proposed a semi-empirical relationship between the ratio of annual evapotranspiration to annual precipitation and the ratio of annual precipitation to annual net radiation. Considerable research has been performed on the www.intechopen.com Budyko curve (Fu, 1981;Milly, 1994;Choudhury, 1999;Wolock and McCabe, 1999;Zhang et al, 2004, Yang et al, 2008a through analyzing the interactions between climate, soils and vegetation in producing annual water balance. A simple water-energy balance equation (Yang et al, 2006 and based on the Budyko hypothesis (1974) has been used for predicting the long-term average evapotranspiration and the inter-annual variability of evapotranspiration.…”