The Budyko functions relate the evaporation ratio E/P (E is evaporation and P precipitation) to the aridity index Φ=Ep/P (Ep is potential evaporation) and are valid on long timescales under steady-state conditions. A new physically based formulation (noted as Moussa–Lhomme, ML) is proposed to extend the Budyko framework under non-steady-state conditions taking into account the change in terrestrial water storage ΔS. The variation in storage amount ΔS is taken as negative when withdrawn from the area at stake and used for evaporation and positive otherwise, when removed from the precipitation and stored in the area. The ML formulation introduces a dimensionless parameter HE=-ΔS/Ep 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 Ep, P and ΔS. For the particular case where the Fu–Zhang equation is used, the ML formulation with ΔS 0 is similar to the analytical solution of Greve et al. (2016) in the standard Budyko space (Ep/P, E/P), a simple relationship existing between their respective parameters. The ML formulation is extended to the space [Ep/ (P-ΔS), E/ (P-ΔS)] 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 Ep/ (P-ΔS) differs: for ΔS0, it is bounded by an upper limit 1/HE in the ML formulation, while it is only bounded by a lower limit in Chen et al.'s (2013) and Du et al.'s (2016) formulations. The ML formulation can also be conducted using the dimensionless parameter HP=-ΔS/P instead of HE, which yields another form of the equations.

Introduction

The Budyko framework has become a simple tool that is widely used within the hydrological community to estimate the evaporation ratio E/P at catchment scale (E is evaporation and P precipitation) as a function of the aridity index Φ=Ep/P (Ep is potential evaporation) through simple mathematical formulations E/P=B1(Φ) and with long-term averages of the variables. Most of the formulations were empirically obtained (e.g., Oldekop, 1911; Turc, 1954; Tixeront, 1964; Budyko, 1974; Choudhury, 1999; Zhang et al., 2001; Zhou et al., 2015), but some of them were analytically derived from simple physical assumptions (Table 1): (i) the one derived by Mezentsev (1955) and then by Yang et al. (2008), which has the same form as the one initially proposed by Turc (1954) (noted hereafter as Turc–Mezentsev); (ii) the one derived by Fu (1981) and reworked by Zhang et al. (2004) (noted hereafter as Fu–Zhang). These two last formulations involve a shape parameter (respectively, λ and ω), which varies with catchment characteristics and vegetation dynamics (Donohue et al., 2007; Yang et al., 2009; Li et al., 2013; Carmona et al., 2014). When its value increases, actual evaporation gets closer to its maximum rate.

The Budyko framework was initially limited to steady-state conditions on long timescales, under the assumption of negligible change in soil water storage and groundwater. 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) and the month (Zhang et al., 2008; Du et al., 2016). However, the water storage variation ΔS 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+ΔS. Table 2 shows some recent formulations of the Budyko framework extended to take into account the change in catchment water storage ΔS. Chen et al. (2013) (used in Fang et al., 2016) and Du et al. (2016) proposed empirical modifications of the Turc–Mezentsev and Fu–Zhang equations, respectively, precipitation P being replaced by the available water supply defined as (P-ΔS), with Du et al. (2016) including the interbasin water transfer into ΔS. Greve et al. (2016) analytically modified the Fu–Zhang equation in the standard Budyko space (Ep/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.

Different expressions for the Budyko curves under steady-state conditions.

Reference Equation E/P=B1(Φ) Budyko (1974) EP=Φtanh1Φ1-exp⁡(-Φ)1/2 Turc (1954) with λ= 2, EP=Φ1+Φλ-1λ Mezentsev (1955), Yang et al. (2008) Fu (1981), EP=1+Φ-1+Φω1ω Zhang et al. (2004) Zhang et al. (2001) EP=1+wΦ1+wΦ+Φ-1 Zhou et al. (2015) EP=Φk1+kΦn1/n

Different expressions for the Budyko curves under non-steady-state conditions.

Reference Steady-state conditions Non-steady-state conditions B1(Φ) Greve et al. (2016) Fu–Zhang EP=1+EpP-1+1-y0κ-1EpPκ1/κ with κ and y0 parameters Chen et al. (2013) Turc–Mezentsev EP-ΔS=1+EpP-ΔS-Φt-λ-1λ with λ and Φt parameters Du et al. (2016) Fu–Zhang EP-ΔS=1+EpP-ΔS-1+EpP-ΔSω+μ1ω with ω and μ parameters

Representation of the change in soil water storage ΔS= (Se-Sb) /Δt for the two cases considered in the paper: ΔS 0 and ΔS 0. Sb and Se are, respectively, the storage at the beginning and the end of the time period Δt.

Upper and lower limits of the feasible domain (in grey) of evaporation in the Turc space (P/Ep, E/Ep) and in the Budyko space (Ep/P, E/P) (water limit in blue, energy limit in green and lower limit in black) when using the non-dimensional parameter HE: (a and b) for steady-state conditions; (c, d, e and f) for non-steady-state conditions with a storage term ΔS (c and d for ΔS 0; and (e) and (f) for ΔS 0).

With the extension of the Budyko framework to non-steady-state conditions being a real challenge, this paper aims to propose a new formulation inferred from a clear physical rationale and compared to other non-steady formulations previously derived. The paper is organized as follows. 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 (Ep/P, E/P) and in the space [Ep/ (P-ΔS), E/ (P-ΔS)]. 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 [Ep/ (P-ΔS), E/ (P-ΔS)].

New generic formulation under non-steady-state conditions Upper and lower limits of the Budyko framework

In the Budyko framework, each catchment is characterized by the three hydrologic variables (P, E and Ep) which are represented in a 2-D space using dimensionless variables equal to the ratio between two of these variables and the third one. In the rest of the paper, following Andréassian et al. (2016), the space defined by (Φ=Ep/P, E/P) is called Budyko space and the one defined by (Φ-1=P/Ep, E/Ep) is called Turc space. For steady-state conditions (ΔS= 0), it should be recalled that any Budyko function B1 defined in the Budyko space (Ep/P, E/P) generates an equivalent function B2 in the Turc space expressed as EEp=B2Φ-1=B1ΦΦ, and that any Budyko function verifies the following conditions under steady-state conditions: (i) E= 0 if P= 0; (ii) EP if PEp (water limit); (iii) EEp if PEp (energy limit); (iv) EEp if P.

First, we present the upper and lower limits in the Turc space under steady-state conditions, when all the water consumed by evaporation comes from the precipitation, the change in water storage ΔS being nil (E=P-Q). Figure 2a represents the variation of maximum and minimum actual evapotranspiration, respectively, Ex and En, as a function of precipitation P with dimensionless variables. The upper solid line represents the dimensionless maximum evaporation rate Ex/Ep: it follows the precipitation up to P/Ep= 1 (the water limit is presented in bold blue on all graphs) and then is limited by potential evaporation Ex/Ep= 1 (the energy limit is in bold green). The lower solid line (in bold black) represents the dimensionless minimum evaporation rate En/Ep which follows the x axis: En/Ep= 0. The feasible domain between the upper and the lower limits is shown in grey. In the Budyko space, we have the following relationships: (i) when evaporation is maximum, for Ep/P 1, Ex/P=Ep/P and for Ep/P 1, Ex/P= 1; (ii) when evaporation is minimum: En/P= 0. The corresponding Budyko non-dimensional graph is shown in Fig. 2b and represents the upper and lower limits of the feasible domain of E/P=B1 (Ep/P).

Under non-steady-state conditions, either a given amount of water ΔS stored in the area at stake participates in the evaporation process (for instance, groundwater depletion for irrigation), or a given amount of the precipitation ΔS is stored in the area (soil water, ground water, reservoirs) following the water balance (E=P-ΔS-Q). As shown in Fig. 1, the storage amount ΔS is taken as negative (ΔS 0) when withdrawn from the area and used for evaporation; it is taken as positive (ΔS 0) when removed from the precipitation and stored in the area. When ΔS is negative, ΔS should be lower than Ep because if ΔSEp, evaporation would be systematically equal to Ep; when ΔS is positive, it should be necessarily lower than P. Consequently, -EpΔSP. The variable ΔS is used in a dimensionless form, either as HE=-ΔS/Ep or HP=-ΔS/P, which are positive when additional water is available for evapotranspiration and negative when water is withdrawn from precipitation. In the following, all the calculations are made with HE (-Φ-1HE 1), but a similar reasoning is conducted using HP (-1 HPΦ) in Appendix A. Taking into account ΔS makes the upper and lower limits of the feasible domain different.

In the Turc space, the case where evaporation is at its maximum value is visualized as the upper limit in Fig. 2c and e (all the available water is used for evaporation). For both cases, ΔS 0 (Fig. 2c) or ΔS 0 (Fig. 2e), we have Ex=P-ΔS if P-ΔSEP and Ex=EP if P-ΔSEP. Written with dimensionless variables, these equations transform into ifPEp1+ΔSEpthenExEp=PEp-ΔSEp=Φ-1+HE,ifPEp1+ΔSEpthenExEp=1. For the minimal value of evapotranspiration En, we have to distinguish two cases depending if ΔS 0 (Fig. 2c) or ΔS 0 (Fig. 2e). ifΔS0thenEnEp=-ΔSEp=HE.ifΔS 0thenEnEp=0

Translating the above equations into the Budyko space (Fig. 2d, f) yields the following for the upper limits: ifEpPEpEp+ΔSthenExP=1-ΔSP=1+HEEpP=1+HEΦ, ifEpPEpEp+ΔSthenExP=EpP=Φ. Equation (5) has two limits: when HE= 0, Ex/P= 1, and when ΔS -Ep, which corresponds to HE= 1, Ex/P (1+ Φ). For the lower limits in the Budyko space we have ifΔS 0thenEnP=-ΔSP=HEEpP=HEΦ,ifΔS 0thenEnP=0. Note that under steady-state conditions, the upper and lower limits are similar in both Turc and Budyko spaces, while this is not the case under non-steady-state conditions. It is also interesting to note that for the negative values of HE the domain of variation of Φ is bounded [0, -1 /HE] and the possible space of the Budyko functions is limited to a triangle (Fig. 2f).

General equations with restricted evaporation

We now examine the case where the evaporation rate is lower than its maximum possible rate. In the Turc space, under non-steady-state conditions (ΔS 0 in Fig. 2c or ΔS 0 in Fig. 2e), Eq. (1) should be transformed to take into account the impact of water storage on the evaporation process. We search a mathematical formulation which transforms the upper and lower limits for the steady-state conditions (Fig. 2a) into the corresponding ones for the non-steady-state conditions (Fig. 2c if ΔS 0 and Fig. 2e if ΔS 0). The mathematical transformation is searched under the form E/Ep=α B2(Φ-1+γ) + β, which combines an x axis translation (γ), a y axis translation (β) and a homothetic transformation (α). This mathematical form is suggested by the way the physical domain of Turc's space is transformed when passing from steady-state conditions to non-steady-state conditions (Fig. 2a, c, e). Note that the reasoning can be conducted either in the Turc or the Budyko space, but the upper and lower limits and the transformation from steady- to non-steady-state conditions are easier to grasp in the Turc space than in the Budyko space. We distinguish the two cases corresponding to ΔS 0 and ΔS 0.

Case <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>≤</mml:mo></mml:math></inline-formula> 0

In the Turc space, the lower limit B2(Φ-1) = 0 in Fig. 2a transforms into B2(Φ-1) =HE in Fig. 2c. Using the mathematical transformation described above, we obtain (α× 0) + β=HE. Following a similar reasoning, the energy limit B2(Φ-1) = 1 transforms into B2(Φ-1) = 1, which yields α + β= 1, and the water limit B2(Φ-1) =Φ-1 transforms into B2(Φ-1) =HE +Φ-1, which yields α (Φ-1 + γ) + β=HE + Φ-1. The resolution of these three equations gives α= 1 - HE, β=HE and γ=Φ-1HE/ (1-HE). Consequently, Eq. (1) should be transformed into EEp=1-HEB2Φ-11-HE+HE. By introducing Eq. (1) into Eq. (8), we obtain the formulation in the Budyko space (Fig. 2d): EP=1-HEΦB2Φ-11-HE+HEΦ=B11-HEΦ+HEΦ. The derivative of Eq. (9) is dEPdΦ=1-HEdB11-HEΦdΦ+HE. Given that dB11-HEΦdΦ=1 for Φ= 0 and dB11-HEΦdΦ=0 when Φ, the derivative dEPdΦ (i.e., the slope of the curve) is equal to 1 for Φ= 0, and when Φ, the derivative tends to HE.

The ML formulation in the Budyko space with the Fu–Zhang relationship Eq. (14a, b) for ω= 1.5 and for different values of HE. The bold lines indicate the upper and lower limits of the feasible domain of evaporation (shown in grey).

Case <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>≥</mml:mo></mml:math></inline-formula> 0

Following the same reasoning as above, the lower limit, the energy limit and the water limit of B2(Φ-1) in the Turc space in Fig. 2a (respectively, 0, 1 and Φ-1) transform, respectively, into 0, 1, and HE + Φ-1 in Fig. 2e. We obtain, respectively, the following three equations: (α× 0) + β= 0, α + β= 1 and α (Φ-1+γ) + β=HE + Φ-1. The resolution gives α= 1, β= 0 and γ=HE. Consequently, Eq. (1) should be transformed into EEp=B2Φ-1+HE. By introducing Eq. (1) into Eq. (11), we obtain the formulation in the Budyko space (Fig. 2f): EP=ΦB2Φ-1+HE=1+HEΦB1Φ1+HEΦ. The derivative of Eq. (12) is dEPdΦ=HEB1Φ1+HEΦ+(1+HEΦ)dB1Φ1+HEΦdΦ. Given that B1Φ1+HEΦ=0 and dB1Φ1+HEΦdΦ=1 for Φ= 0, the derivative dEPdΦ is equal to 1 for Φ= 0. When Φ-1/HE, B1Φ1+HEΦ=1 and dB1Φ1+HEΦdΦ=0, the derivative dEPdΦ tends to HE.

In the following, these new generic formulae (Eqs. 8 and 9 for ΔS 0 and Eqs. 11 and 12 for ΔS 0) are called ML formulations (ML stands for Moussa–Lhomme).

Application

Any Budyko equation B1(Φ) from Table 1 can be used in Eqs. (9) and (12) as detailed in Table S1 in the Supplement. It is worth noting that both the Turc–Mezentsev and Fu–Zhang functions, which are obtained from the resolution of a Pfaffian differential equation, have the following remarkable simple property: F(1/x) =F(x) /x. This means that the same mathematical expression is valid for B1 and B2: B1=B2. 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 B1 by Fu–Zhang's equation, in Eq. (9) for ΔS 0 and in Eq. (12) for ΔS 0, in the Budyko space, gives ifΔS 0thenEP=1+Φ-1+1-HEωΦω1ω, ifΔS 0thenEP=1+1+HEΦ-1+HEΦω+Φω1ω. For Φ= 0, and in both cases ΔS 0 and ΔS 0, we have E/P= 0. However, the upper limits of Φ differ: for ΔS 0, when Φ, E/P, while for ΔS 0 the maximum value of Φ is -1/HE and corresponds to E/P= 0. Figure 3 shows some examples of the shape of the ML formulation in the Budyko space (Eq. 14a, b) for ω= 1.5 and different values of HE. Note that for the particular and unlikely case when HE-, upper and lower limits are reduced to the point (Ep/P= 0, E/P= 0). For HE= 0, we obtain the curves corresponding to steady-state conditions, while for HE= 1, upper and lower limits are superimposed, and the domain is restricted to the 1 : 1 line. We can easily verify that all functions in Table S1 of the Supplement give similar results.

The ML formulation with the Fu–Zhang Eq. (21a, b) in the space [Ep/ (P-ΔS), E/ (P-ΔS)] for ω= 1.5 and four values of HE. For HE 0, all curves have a common upper end at Φ= 1 /HE corresponding to E/ (P-ΔS) = 1. The bold lines indicate the upper and lower limits of the feasible domain shown in grey. For HE 0, the curve is similar to the one under steady-state conditions.

The ML formulation in the space [<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula>), <inline-formula><mml:math display="inline"><mml:mi>E</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula>)]

As mentioned in the introduction, some authors (Chen et al., 2013; Du et al., 2016) have dealt with the non-steady conditions by modifying the Budyko reference space and replacing the precipitation P by P-ΔS. Hereafter, we develop the ML formulations in this new space. The upper limits of the ML formulations can be obtained by transforming Eqs. (5) and (6) defined in the Budyko space. We get, respectively, ifEpP-ΔS1thenExP-ΔS=1,ifEpP-ΔS<1thenExP-ΔS=EpP-ΔS. The lower limits are obtained from Eq. (7a, b): ifΔS 0thenEnP-ΔS=-ΔSP-ΔS=HEEpP-ΔS,ifΔS 0thenEnP-ΔS=0. In the new space, we put Φ=EpP-ΔS=Φ1+HEΦorΦ=Φ1-HEΦ. Consequently, the relationship between E/ (P-ΔS), Φ and E/P is given by EP-ΔS=EPPP-ΔS=11+HEΦEP=1-HEΦEP. Inserting Eqs. (9) and (12) into Eq. (19) and expressing Φ as a function of Φ (Eq. 18) led to the ML formulation in the new space: ifΔS 0thenEP-ΔS=11+HEΦB11-HEΦ+HEΦ=1-HEΦB11-HEΦ1-HEΦ+HEΦ, ifΔS 0thenEP-ΔS=11+HEΦ1+HEΦB1Φ1+HEΦ=B1Φ. Note that for ΔS 0, E/ (P-ΔS) =B1(Φ) is independent of HE and is identical to the steady-state condition HE= 0. This is explained by the fact that the stored water ΔS being initially subtracted to the precipitation P, it does not participate in the evaporation process and consequently has no impact on the ratio E/ (P-ΔS). For ΔS 0, and for Φ= 0, i.e., P, we have Φ= 0, B1= 0 and E/ (P-ΔS) = 0. When Φ, which corresponds to P 0, we have Φ= 1 /HE, B1= 1, and E/ (P-ΔS)  1.

Any Budyko formulation B1 in Table 1 can be used with Eq. (20a, b), as shown in Table S2 of the Supplement. When the Fu–Zhang equation is used, Eq. (20a, b) become ifΔS0thenEP-ΔS=1+1-HEΦ-1-HEΦω+1-HEωΦω1/ω, ifΔS0thenEP-ΔS=1+Φ-1+Φω1ω. Figure 4 shows the ML formulation (Eq. 21a, b) in the space [Ep/ (P-ΔS), E/ (P-ΔS)] for ω= 1.5 and different values of HE. For HE= 0, we retrieve the original Fu–Zhang equation and when ω= 1, we can easily verify that Eq. (21a, b) are equal to the lower limit of the domain E/ (P-ΔS) =HE Ep/ (P-ΔS) when ΔS 0, and E/ (P-ΔS) = 0 when ΔS 0.

The ML formulation using the dimensionless parameter <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mi mathvariant="normal">P</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>

A mathematical development, similar to the one of Sect. 2.1, 2.2 and 2.3, is conducted in Appendix A using the dimensionless parameter HP=-ΔS/P=HEΦ (instead of HE=- ΔS/EP) and yields another form of the ML formulation. Equivalent mathematical representations are obtained for ΔS 0 and ΔS 0 in the different spaces explored in Sect. 2.1, 2.2 and 2.3. Figures S1, S2 and S3 in the Supplement obtained with the parameter HP correspond, respectively, to Figs. 2, 3 and 4 obtained with HE. Similarly, Tables S3 and S4 in the Supplement (obtained with HP) correspond to Tables S1 and S2 in the Supplement (obtained with HE): they give the ML formulation applied to the different Budyko curves of Table 1 in the standard Budyko space (Ep/P, E/P) and in the space [Ep/ (P-ΔS), E/ (P-ΔS)]. Significant differences appear concerning the mathematical equations and the shape of the feasible domain defined by its upper and lower limits. This is due to the fact that using HE or HP corresponds to different sets of data and different functional representations. Both approaches (HE or HP) can be used. When storage water contributes to enhancing evaporation (ΔS 0), ΔS is bounded by potential evaporation EP and consequently represents a given percentage of EP. Hence, it is more convenient to use HE=-ΔS/Ep instead of HP=-ΔS/P, because HE lies in the range [0,1] which is not the case for HP. Conversely, when precipitation water contributes to increase soil water storage (ΔS 0), ΔS is bounded by P and represents a percentage of precipitation P. Consequently, using HP is more convenient because HP lies in the range [-1,0]. Moreover, in order to keep the parameter in the range [0,1], HP=-HP could be preferred.

Comparing the new formulation with other formulae from the literature In the standard Budyko space (<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>, <inline-formula><mml:math display="inline"><mml:mi>E</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula>)

When evapotranspiration exceeds precipitation (corresponding herein to the case ΔS 0), Greve et al. (2016) analytically developed a Budyko-type equation where the water storage is taken into account through a parameter y0 (0y01) introduced into the Fu–Zhang formulation (Table 2). In the Budyko space, this equation is written (Greve et al., 2016; Eq. 9) as EP=1+Φ-1+1-y0κ-1Φκ1/κ. They used the shape parameter κ to avoid confusion with the traditional ω of Fu–Zhang equation. Despite different physical and mathematical backgrounds, Eqs. (14a) and (22) are exactly identical and a simple relationship between HE and y0 can be easily obtained. Equating Eqs. (14a) and (22) with ω=κ yields HE=1-1-y0ω-1ω. The relationship between y0 and HE is independent from Φ. It is shown in Fig. 5 for different values of ω. For a given value of ω, we have HE < y0. For ω= 1, we have HE= 0, and when ω, we have HE=y0.

Relationship (Eq. 23) between the parameter HE of the ML formulation (Eq. 14a) and the parameter y0 of the Greve et al. (2016) (Eq. 22) for different values of ω with ω=κ.

Example showing the similarity of the ML formulation Eq. (14a) and the equation of Greve et al. (2016) Eq. (22) (with ω=κ= 2) for different values of y0; the corresponding values of HE are calculated using Eq. (23).

The derivative of Eq. (22) gives dEPdΦ=1-1-y0κ-1Φκ-11+1-y0κ-1Φκ1-κκ. For Φ= 0, the derivative is equal to 1, and when Φ, the derivative tends to a value noted as m by Greve et al. (2016; Eq. 12): m=1-1-y0κ-1κ. The value of the derivative (slope of the curve) is the same in both ML and Greve et al.'s (2016) formulations: for Φ= 0 the derivative is equal to 1, and when Φ we have m=HE (assuming ω=κ). Greve et al. (2016; Sect. 4) show that y0 is the maximum value of m reached when ω. Hence, substituting in Eq. (22) y0 by its value inferred from Eq. (23) yields an equation identical to that obtained from the ML formulation (Eq. 14a).

Figure 6 compares the ML formulation Eq. (14a) with Greve et al.'s (2016) analytical solution Eq. (22) for ω=κ= 2 and different values of y0 (0, 0.2, 0.4, 0.6, 0.8 and 1). The corresponding values of HE (respectively, 0, 0.106, 0.225, 0.367, 0.553 and 1) are calculated using Eq. (23). The new ML formulation with ω=κ, and only for ΔS 0, gives exactly the same curves as those obtained by Greve et al. (2016). Both formulations are identical and have the same upper and lower limits. Greve et al. (2016), however, did not mention the lower limit and limited the reasoning to positive values of y0. Moreover, the case of ΔS 0 is not considered by Greve et al. (2016).

In the space [<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula>), <inline-formula><mml:math display="inline"><mml:mi>E</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula>)]

The formulations proposed by Chen et al. (2013) and Du et al. (2016) in the space [Ep/(P-ΔS),(E/(P-ΔS)] are essentially empirical. The Chen et al. (2013) function (Table 2) is derived from the Turc–Mezentsev equation and written as EP-ΔS=1+EpP-ΔS-Φt-λ-1λ. An additional parameter Φt is empirically introduced in order “to characterize the possible non-zero lower bound of the seasonal aridity index”; this parameter causes a shift of the curve E/ (P-ΔS) along the horizontal axis such as for Ep/(P-ΔS)=Φt, we have E/ (P-ΔS) = 0. The derivative of Eq. (26) when Ep/ (P-ΔS)  is equal to 0. Similarly, the Du et al. (2016) function (Table 2) is an empirical modification of Fu–Zhang equation (Fu, 1981; Zhang et al., 2004) written as EP-ΔS=1+EpP-ΔS-1+EpP-ΔSω+μ1ω. A supplementary parameter, noted here as μ (> -1), is added to modify the lower bound of the aridity index EP/ (P-ΔS). The parameter μ plays a similar role as Φt in Eq. (26). For μ= 0, Eq. (27) takes the original form of Fu–Zhang equation, (P-ΔS) replacing P. When μ becomes positive, the lower end of the curve E/ (P-ΔS) shifts to the right. The function E/ (P-ΔS) in Eq. (27) is equal to zero for the particular value of Ep/ (P-ΔS) =Φd, such as 1+Φdω=1+Φdω+μ. Greve et al.'s (2016) formulation can be also written in the space [Ep/ (P-ΔS), E/ (P-ΔS)]. Inserting Eq. (22) into Eq. (20a) and expressing Φ as a function of Φ (Eq. 18) leads to EP-ΔS=1+1-HEΦ-1-HEΦκ+1-y0κ-1Φκ1/κ. It can be mathematically shown that expressing (1-y0) in Eq. (29) as a function of HE by inverting Eq. (23) (assuming ω=κ) leads to the exact ML formulation of Eq. (21a). It is a direct consequence of the similarity of both formulations. Therefore, similar curves to those shown in Fig. 4 for the ML formulation with HE 0 are obtained with Greve et al.'s (2016) formulation.

For ΔS 0 (corresponding to HE 0; Fig. 4), the three formulations (Chen et al.'s, 2013, Du et al.'s 2016, and ML) have similar upper and lower limits. For ΔS 0, Fig. 7 shows an example of the curves obtained with Du et al.'s (2016) equation (ω= 1.5) and Chen et al.'s (2013) equation with λ= 0.78 (such as λ=ω-0.72 from Yang et al., 2008) and with Φt=Φd= 0.5 (corresponding to μ= 0.484 from Eq. 28). Both Chen et al.'s (2013) and Du et al.'s (2016) formulations are compared to the ML formulation using the Fu–Zhang Eq. (14a) with HE=+ 0.25. The ML and Greve et al. (2016) formulations are exactly identical if κ=ω= 1.5 and y0= 0.578 calculated from Eq. (23) for HE=+0.25. The four formulations have similar upper limits but the lower limits are different. Both Chen et al.'s (2013) and Du et al.'s (2016) formulations have the x axis as the lower limit and E/ (P-ΔS) tends to 1 when Φ=Ep/ (P-ΔS) , while in the ML formulation with ΔS 0 (Fig. 2a) the feasible domain is a triangle, the domain of variation of Φ being limited by 0 and 1 /HE.

In the space [Ep/ (P-ΔS), E/ (P-ΔS)], an example comparing the three formulations: Du et al. (2016) with ω= 1.5 and Φd= 0.5; Chen et al. (2013) with λ=ω-0.72= 0.78 and Φt=Φd= 0.5; the ML formulation for ΔS 0 (Eq. 14a) with ω= 1.5 and HE= 0.25 (identical to Greve et al. 2016 formulation). The feasible domain of the ML formulation in dark grey is superimposed over the domains of both Chen et al. (2013) and Du et al. (2016) in light grey.

Discussion

All four formulations (ML; Greve et al., 2016; Chen et al., 2013; and Du et al., 2016) have two parameters each: one for the shape of the curve and another for its shift due to non-steady conditions: ω and HE for the ML formulation (with the Fu–Zhang function), κ and y0 for Greve et al. (2016), λ and Φt for Chen et al. (2013), ω and μ for Du et al. (2016). If HE=y0=Φt=μ= 0, the four formulations are identical. For ΔS 0, the ML formulation with the Fu–Zhang equation (Eq. 14a) is identical to the one of Greve et al. (2016) in the Budyko space and also in the [Ep/ (P-ΔS), E/ (P-ΔS)] space, provided the shape parameters are assumed to be identical (ω=κ) (a simple relationship is established between HE and the corresponding parameter y0). Despite similar upper limits, the ML and Greve et al. (2016) formulations behave very differently from those of Chen et al. (2013) and Du et al. (2016) in the space [Ep/ (P-ΔS), E/ (P-ΔS)]. The ML formulation is different for ΔS 0 and ΔS 0, while those of Chen et al.'s (2013) and Du et al.'s (2016) do not distinguish the two cases ΔS 0 and ΔS 0. All the formulations have the same upper limits, but the domain of variation of Φ differs: respectively, [0, 1 /HE] when ΔS 0 and [0, ] when ΔS 0 for the ML formulation, [Φt,] for Chen et al. (2013) and [Φd,] for Du et al. (2016) The lower end of the curve E/ (P-ΔS) corresponds, respectively, to (0, 0), (Φt, 0) and (Φd, 0) and the upper end to (1 /HE, 1) when ΔS 0 and (, 1) when ΔS 0 for the ML formulation, (, 1) for the other two. Moreover, the ML formulation for ΔS 0 is reduced to a simple relationship E/ (P-ΔS) =B1 (Φ) and is independent of HE.

It is worth noting that for ΔS 0 the limits of Chen et al. (2013) and Du et al. (2016) functions are not completely sound from a strict physical standpoint: for very high precipitation, when PEp, Φ and Φ should logically tend to zero and not to Φt and Φd; similarly, when P 0, i.e., Φ, it is physically logical that ΦEp/ (-ΔS) = 1 /HE, as predicted by our Eq. (20a). This tends to prove that the ML formulation, corroborated by the Greve et al. (2016) formulation, is physically more correct. Additionally, at simple glimpse, we note that the ML curves could be easily adjusted to the set of experimental points shown in Chen et al. (2013; Figs. 2 and 9) and in Du et al. (2016; Figs. 8 and 9).

Conclusion

The ML formulations constitute a general mathematical framework which allows any standard Budyko function developed at catchment scale under steady-state conditions (Table 1) to be extended to non-steady conditions (Table S1 in the Supplement). They take into account the change in catchment water storage ΔS through a dimensionless parameter HE=-ΔS/Ep and the formulation differs according to the sign of ΔS (Eqs. 8 and 9 for ΔS 0 and Eqs. 11 and 12 for ΔS 0). Applications can be conducted at various time steps (yearly, seasonal or monthly) both in the Turc space (P/Ep, E/Ep) and in the standard Budyko space (Ep/P, E/P), with the only data required to obtain E being Ep, P and ΔS.

The new formulations are inferred from an evaluation of the feasible domain of evaporation in the Turc space, adjusted for the case where additional (ΔS 0) or restricted (ΔS 0) water is available for evaporation, and then transformed in the Budyko space. For ΔS= 0, the ML formulations return the original equations under steady-state conditions, with similar upper and lower limits in both spaces. Under non-steady-state conditions, however, the upper and lower limits of the feasible domain differ when using either the Turc or the Budyko space. The ML formulations can be extended to the [Ep/ (P-ΔS), E/ (P-ΔS)] space (Eq. 20a, b, Fig. 4). They can also be conducted using the dimensionless parameter HP=-ΔS/P instead of HE, which yields another form of the equations (Appendix A and the Supplement). It is shown that the ML formulation with ΔS 0 is identical to the analytical solution of Greve et al. (2016) in the standard Budyko space, a simple relationship existing between their respective parameters. On the other hand, the new formulation differs from those of Chen et al. (2013) and Du et al. (2016) in the space [Ep/ (P-ΔS), E/ (P-ΔS)].

Scaling <inline-formula><mml:math display="inline"><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula> by <inline-formula><mml:math display="inline"><mml:mi>P</mml:mi></mml:math></inline-formula> instead of <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>

Appendix A presents the set of equations when scaling the change in soil water storage ΔS by precipitation P instead of potential evaporation Ep, i.e., using HP=-ΔS/P=HEΦ (-1 HPΦ) instead of HE=-ΔS/Ep (-Φ-1HE 1).

Upper and lower limits of the Budyko framework

In the Turc space, the upper limits of evapotranspiration Ex/Ep are obtained from Eqs. (2) and (3): ifPEp1+ΔSEpthenExEp=PEp-ΔSEp=1+HPΦ-1,ifPEp1+ΔSEpthenExEp=1, and the lower limits of evapotranspiration En/Ep from Eq. (4a, b): ifΔS 0thenEnEp=-ΔSEp=HPΦ-1,ifΔS 0thenEnEp=0.

The translation in the Budyko space yields the following for the upper limits: ifEpPEpEp+ΔSthenExP=1-ΔSP=1+HP,ifEpPEpEp+ΔSthenExP=EpP=Φ, and the following for the lower limits: ifΔS 0thenEnP=-ΔSP=HEEpP=HEΦ=HP, ifΔS 0thenEnP=0.

In the Supplement, Fig. S1 shows the upper and lower limits of the feasible domain of evaporation in the Turc and Budyko spaces, drawn with the parameter HP=-ΔS/P. Figure S1 in the Supplement corresponds to Fig. 2 obtained with HE=-ΔS/Ep.

General equations with restricted evaporation

We distinguish two cases: ΔS 0 and ΔS 0. Substituting HE by HP/Φ in Eqs. (8), (9), (11) and (12) we obtain (if ΔS 0 in the Turc space) EEp=1-HPΦ-1B2Φ-11-HPΦ-1+HPΦ-1, then in the Budyko space EP=B1Φ-HP+HP. If ΔS 0 in the Turc space EEp=B21+HPΦ-1, then in the Budyko space EP=1+HPB1Φ1+HP. Replacing B1 by Fu–Zhang's equation, in Eq. (A8) for ΔS 0 and in Eq. (A10) for ΔS 0, gives the following in the Budyko space: ifΔS 0thenEP=1+Φ-1+Φ-HPω1ω,ifΔS 0thenEP=1+Φ+HP-1+HPω+Φω1ω. In the Supplement, Fig. S2 shows an example of the ML formulation (Eq. A11a, b) in the Budyko space obtained with the parameter HP=-ΔS/P. It corresponds to Fig. 3 obtained with HE=-ΔS/Ep. Table S3 gives the ML formulation applied to the different Budyko curves of Table 1 with the parameter HP (Eqs. A8 and A10). It corresponds to Table S1 in the Supplement obtained with HE.

The ML formulation in the space [<inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi mathvariant="normal">p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula>), <inline-formula><mml:math display="inline"><mml:mi>E</mml:mi></mml:math></inline-formula> <inline-formula><mml:math display="inline"><mml:mo>/</mml:mo></mml:math></inline-formula> (<inline-formula><mml:math display="inline"><mml:mrow><mml:mi>P</mml:mi><mml:mo>-</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:math></inline-formula>)]

Equations (15), (16), (17a) and (17b) yield the following for the upper limits: ifEpP-ΔS1thenExP-ΔS=1,ifEpP-ΔS1thenExP-ΔS=EpP-ΔS, and the following for the lower limits: ifΔS 0thenEnP-ΔS=-ΔSP-ΔS=HEEpP-ΔS=HPHP+1, ifΔS 0thenEnP-ΔS=0. In the new space, [Ep/(P-ΔS), E/ (P-ΔS)], we put Φ=EpP-ΔS=Φ1+HPorΦ=1+HPΦ. Consequently, the relationship between E/ (P-ΔS) and E/P is given by EP-ΔS=EPPP-ΔS=11+HpEP.

Replacing HE by HP/Φ in Eq. (20a, b) we obtain ifΔS 0thenEP-ΔS=11+HPB1Φ-HP+HP=11+HPB11+HPΦ-HP+HP1+HP, ifΔS 0thenEP-ΔS=11+HP1+HPB1Φ1+HP=B1Φ. Using the Fu–Zhang equation for B1 we get ifΔS 0thenEP-ΔS=1+Φ-HP1+HP-11+HPω+Φ-HP1+HPω1/ω, ifΔS 0thenEP-ΔS=1+Φ-1+Φω1ω. In the Supplement, Fig. S3 shows an example of the ML formulation (Eq. A18a, b) in the space [Ep/ (P-ΔS), E/ (P-ΔS)] obtained with the parameter HP=-ΔS/P. It corresponds to Fig. 4 obtained with HE=-ΔS/Ep. Table S4 in the Supplement gives the ML formulation applied to the different Budyko curves of Table 1 in the space [Ep/ (P-ΔS), E/ (P-ΔS)] with the parameter HP (Eq. A17a and b). It corresponds to Table S2 in the Supplement obtained with HE.

List of symbols

B1(Φ) Relationship between E/P and Φ in the Budyko space (Ep/P, E/P) such as E/P=B1(Φ) [–] B2(Φ-1) Relationship between E/Ep and Φ-1=P/Ep in the Turc space (P/Ep, E/Ep) such as E/Ep=B2(Φ-1) [–] E actual evaporation [LT-1] En Lower limit of the feasible domain of evaporation [LT-1] Ep Potential evaporation [LT-1] Ex Upper limit of the feasible domain of evaporation [LT-1] HE -ΔS/Ep (-P/EpHE 1) [–] HP -ΔS/P (-1 HPEp/P) [–] m Slope of the equation of Greve et al. (2016) when Φ [–] ML New formulations: Eqs. (8) and (9) for ΔS 0 and Eqs. (11) and (12) for ΔS 0 (stands for Moussa–Lhomme) P Precipitation [LT-1] Q Runoff [LT-1] y0 Parameter in the Greve et al. (2016) equation accounting for non-steady-state conditions (0 y0 1) [–] κ Shape parameter in the Greve et al. (2016) equation corresponding to ω in the Fu–Zhang equation [–] ΔS Water storage variation [LT-1] λ Shape parameter in the Turc–Mezentsev equation (λ > 0) [–] μ Parameter in the Du et al. (2013) equation [–] Φ Aridity index (Φ=Ep/P) [–] Φd Aridity index threshold in the Du et al. (2016) equation corresponding to E/ (P-ΔS) = 0 [–] Φt Aridity index threshold in the Chen et al. (2013) equation [–] Φ Ep/(P-ΔS) [–] ω Shape parameter of the Fu–Zhang equation (ω > 1) [–]