New interpretation of the role of water balance in an extended Budyko hypothesis in arid regions

Abstract

Abstract. The Budyko hypothesis (BH) is an effective approach to investigating long-term water balance at large basin scale under steady state. The assumption of steady state prevents applications of the BH to basins, which is unclosed, or with significant variations in root zone water storage, i.e., under unsteady state, such as in extremely arid regions. In this study, we choose the Heihe River basin (HRB) in China, an extremely arid inland basin, as the study area. We firstly use a calibrated and then validated monthly water balance model, i.e., the abcd model, to quantitatively determine annual and monthly variations of water balance for the sub-basins and the whole catchment of the HRB, and find that the roles of root zone water storage change and that of inflow from upper sub-basins in monthly water balance are significant. With the recognition of the inflow water from other regions and the root zone water storage change as additional possible water sources to evapotranspiration in unclosed basins, we further define the equivalent precipitation (P e ) to include local precipitation, inflow water and root zone water storage change as the water supply in the Budyko framework. With the newly defined water supply, the Budyko curve can successfully describe the relationship between the evapotranspiration ratio and the aridity index at both annual and monthly timescales, whilst it fails when only the local precipitation being considered. Adding to that, we develop a new Fu-type Budyko equation with two non-dimensional parameters (ω and λ) based on the deviation of Fu's equation. Over the annual timescale, the new Fu-type Budyko equation developed here has more or less identical performance to Fu's original equation for the sub-basins and the whole catchment. However, over the monthly timescale, due to large seasonality of root zone water storage and inflow water, the new Fu-type Budyko equation generally performs better than Fu's original equation. The new Fu-type Budyko equation (ω and λ) developed here enables one to apply the BH to interpret regional water balance over extremely dry environments under unsteady state (e.g., unclosed basins or sub-annual timescales).

The Budyko functions under non-steady-state conditions

“…S b and S e are, respectively, the storage at the beginning and the end of the time period t. and the month (Zhang et al, 2008;Du et al, 2016). However, the water storage variation S cannot be considered as negligible when dealing with these finer timescales or for unclosed basins (e.g., soil, groundwater, reservoir, snow, interbasin water transfer, irrigation; Jaramillo and Destouni, 2015).…”

“…In these cases, the catchment is considered to be under non-steady-state conditions ( Fig. 1) and the basin water balance should be written as P = E + Q + S. Table 2 shows some recent formulations of the Budyko framework extended to take into account the change in catchment water storage S. Chen et al (2013) (used in Fang et al, 2016 and Du et al (2016) proposed empirical modifications of the TurcMezentsev and Fu-Zhang equations, respectively, precipitation P being replaced by the available water supply defined as (P − S), with Du et al (2016) including the interbasin water transfer into S. Greve et al (2016) analytically modified the Fu-Zhang equation in the standard Budyko space (E p / P , E / P ) introducing an additional parameter, whereas Wang and Zhou (2016) proposed in the same Budyko space a formulation issued from the hydrological ABCD model (Alley, 1984), but with two additional parameters.…”

“…Hydrological processes leading to changes in water storage are not represented and the catchment is considered closed without any anthropogenic intervention: precipitation is the only input and evaporation and runoff Q the only outputs (P = E + Q). Recently, the Budyko framework has been downscaled to the year (Istanbulluoglu et al, 2012;Wang, 2012;Carmona et al, 2014;Du et al, 2016), the season (Gentine et al, 2012;Chen et al, 2013;Greve et al, 2016) Published by Copernicus Publications on behalf of the European Geosciences Union. 4868 R. Moussa and J.-P. Lhomme: The Budyko functions under non-steady-state conditions Table 1.…”

“…The ML formulation is extended to the space [E p / (P − S), E / (P − S)] and compared to the formulations of Chen et al (2013) and Du et al (2016). The ML (or Greve et al, 2016) feasible domain has a similar upper limit to that of Chen et al (2013) and Du et al (2016), but its lower boundary is different. Moreover, the domain of variation of E p / (P − S) differs: for S ≤ 0, it is bounded by an upper limit 1 / H E in the ML formulation, while it is only bounded by a lower limit in Chen et al.…”

“…It represents a generic framework, easy to use at various time steps (year, season or month), with the only data required being E p , P and S. For the particular case where the Fu-Zhang equation is used, the ML formulation with S ≤ 0 is similar to the analytical solution of Greve et al (2016) in the standard Budyko space (E p / P , E / P ), a simple relationship existing between their respective parameters. The ML formulation is extended to the space [E p / (P − S), E / (P − S)] and compared to the formulations of Chen et al (2013) and Du et al (2016). The ML (or Greve et al, 2016) feasible domain has a similar upper limit to that of Chen et al (2013) and Du et al (2016), but its lower boundary is different.…”