A two-parameter Budyko function to represent conditions under which evapotranspiration exceeds precipitation

Abstract

Abstract. A comprehensive assessment of the partitioning of precipitation (P ) into evapotranspiration (E) and runoff (Q) is of major importance for a wide range of socio-economic sectors. For climatological averages, the Budyko framework provides a simple first-order relationship to estimate water availability represented by the ratio E / P as a function of the aridity index (E p /P , with E p denoting potential evaporation). However, the Budyko framework is limited to steadystate conditions, being a result of assuming negligible storage change in the land-water balance. Processes leading to changes in the terrestrial water storage at any spatial and/or temporal scale are hence not represented. Here we propose an analytically derived modification of the Budyko framework including a new parameter explicitly representing additional water available to evapotranspiration besides instantaneous precipitation. The modified framework is comprehensively analyzed, showing that the additional parameter leads to a rotation of the original water supply limit. We further evaluate the new formulation in an example application at mean seasonal timescales, showing that the extended framework is able to represent conditions in which monthly to annual evapotranspiration exceeds monthly to annual precipitation.

The Budyko functions under non-steady-state conditions

“…First, we present the new formulation under non-steady-state conditions: its upper and lower limits, its generic equations under restricted evaporation in the Budyko space (E p / P , E / P ) and in the space [E p / (P − S), E / (P − S)]. Second, we compare the new formulation to the analytical solution of Greve et al (2016) in the standard Budyko space and to the formulations of Chen et al (2013) and Du et al (2016) in the space [E p / (P − S), E / (P − S)].…”

“…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.…”

“…Both Turc-Mezentsev and Fu-Zhang functions have similar shapes, and a simple linear relationship was established by Yang et al (2008) between their parameters (see Table 1): ω = λ + 0.72. The ML formulation is used hereafter with the Fu-Zhang function (Table 1) for comparison with the analytical solution of Greve et al (2016) based upon the same function. Replacing B 1 by in Eq.…”

“…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 introduces a dimensionless parameter H E = − S / E p and can be applied with any Budyko function. 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).…”