The Budyko functions under non-steady state conditions

The Budyko functions relate the evaporation ratio E/P (E is evaporation and P precipitation) to the aridity index  = Ep/P (Ep is potential evaporation) and are valid on long timescales under steady state conditions. A new physically based formulation (noted ML) is proposed to extend the Budyko framework under non-steady state conditions taking into account the change in terrestrial water storage S. The variation in storage amount S is taken as negative when withdrawn from the 10 area at stake and used for evaporation and positive otherwise, when removed from the precipitation and stored in the area.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 = -S/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), the only data required being Ep, P and ΔS. 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 (Ep/P, E/P), a simple relationship existing between 15 their respective parameters. The ML formulation is extended to the space [(Ep/(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.) feasible domain has similar upper limit to that of Chen et al. and Du et al., but its lower boundary is different. Moreover, the domain of variation of Ep/(P-S) differs: for S 0 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 and Du et al.’s formulations. The ML formulation can also be conducted using the dimensionless parameter HP = -ΔS/P 20 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 = E p / P (E p is potential evaporation) through simple mathematical formulations E / P = B 1 ( ) and with longterm 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 steadystate 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) Published by Copernicus Publications on behalf of the European Geosciences Union.

Reference
Equation E / P = B 1 ( ) Budyko ( 1974) Turc (1954) with λ = 2, E P = 1 + λ − 1 λ Mezentsev (1955), Yang et al. (2008) Fu ( 1981), 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 Turc-Mezentsev 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.
With the extension of the Budyko framework to nonsteady-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 (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)].
2 New generic formulation under non-steady-state conditions 2.1 Upper and lower limits of the Budyko framework In the Budyko framework, each catchment is characterized by the three hydrologic variables (P , E and E p ) 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 ( = E p / P , E / P ) is called Budyko space and the one defined by ( −1 = P / E p , E / E p ) is called Turc space.For steady-state conditions ( S = 0), it should be recalled that any Budyko function B 1 defined in the Budyko space (E p / P , E / P ) generates an equivalent function B 2 in the Turc space expressed as and that any Budyko function verifies the following conditions under steady-state conditions: (i) 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 S being nil (E = P −Q). Figure 2a represents the variation of maximum and minimum actual evapotranspiration, respectively, E x and E n , as a function of precipitation P with dimensionless variables.The upper solid line represents the dimensionless maximum evaporation rate E x / E p : it follows the precipitation up to P / E p = 1 (the water limit is presented in bold blue on all graphs) and then is limited by potential evaporation E x / E p = 1 (the energy limit is in bold green).The lower solid line (in bold black) represents the dimensionless minimum evaporation rate E n / E p which follows the x axis: E n / E p = 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 E p / P ≤ 1, E x / P = E p / P and for E p / P ≥ 1, E x / P = 1; (ii) when evaporation is minimum: E n / 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 = B 1 (E p / P ).

Steady-state conditions
Non-steady-state conditions Upper and lower limits of the feasible domain (in grey) of evaporation in the Turc space (P / E p , E / E p ) and in the Budyko space (E p / P , E / P ) (water limit in blue, energy limit in green and lower limit in black) when using the non-dimensional parameter H E : (a and b) for steady-state conditions; (c, d, e and f) for non-steady-state conditions with a storage term S (c and d for S ≤ 0; and (e) and (f) for S ≥ 0).
evaporation process (for instance, groundwater depletion for irrigation), or a given amount of the precipitation S is stored in the area (soil water, ground water, reservoirs) following the water balance (E = P − S − Q).As shown in Fig. 1, the storage amount S is taken as negative ( S ≤ 0) when withdrawn from the area and used for evaporation; it is taken as positive ( S ≥ 0) when removed from the precipitation and stored in the area.When S is negative, | S| should be lower than E p because if | S| ≥ E p , evaporation would be systematically equal to E p ; when S is positive, it should be necessarily lower than P .Consequently, −E p ≤ S ≤ P .The variable S is used in a dimensionless form, either as H E = − S / E p or H P = − S / 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 H E (− −1 ≤ H E ≤ 1), but a similar reasoning is conducted using Taking into account S 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, S ≤ 0 (Fig. 2c) or S ≥ 0 (Fig. 2e), we have if For the minimal value of evapotranspiration E n , we have to distinguish two cases depending if S ≤ 0 (Fig. 2c) or S ≥ 0 (Fig. 2e). if Translating the above equations into the Budyko space (Fig. 2d, f) yields the following for the upper limits: if Equation ( 5) has two limits: when H E = 0, E x / P = 1, and when S → −E p , which corresponds to H E = 1, E x / P → (1+ ).For the lower limits in the Budyko space we have if S ≤ 0 then if S ≥ 0 then 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 H E the domain of variation of is bounded [0, −1 / H E ] 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 ( S ≤ 0 in Fig. 2c or S ≥ 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 nonsteady-state conditions (Fig. 2c if S ≤ 0 and Fig. 2e if S ≥ 0).The mathematical transformation is searched under the form E / E p = α B 2 ( −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 nonsteady-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 S ≤ 0 and S ≥ 0.

Application
Any Budyko equation B 1 ( ) 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 B 1 and B 2 : 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 Fu-Zhang's equation, in Eq. ( 9) for S ≤ 0 and in Eq. ( 12) for S ≥ 0, in the Budyko space, gives if S ≤ 0 then For = 0, and in both cases S ≤ 0 and S ≥ 0, we have E / P = 0.However, the upper limits of differ: for S ≤ 0, when → ∞, E / P → ∞, while for S ≥ 0 the maximum value of is −1/H E 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 H E .Note that for the particular and unlikely case when H E → −∞, upper and lower limits are reduced to the point (E p / P = 0, E / P = 0).For H E = 0, we obtain the curves corresponding to steady-state conditions, while for H E = 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.

2.3
The ML formulation in the space [E p / (P − S), E / (P − S)] 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 − S. 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, if www.hydrol-earth-syst-sci.net/20/4867/2016/ Hydrol.Earth Syst.Sci., 20, 4867-4879, 2016 The lower limits are obtained from Eq. (7a, b): if S ≥ 0 then In the new space, we put Consequently, the relationship between E / (P − S), and E / P is given by Inserting Eqs. ( 9) and ( 12) into Eq.( 19) and expressing as a function of (Eq.18) led to the ML formulation in the new space: if S ≥ 0 then Note that for S ≥ 0, E / (P − S) = B 1 ( ) is independent of H E and is identical to the steady-state condition H E = 0.This is explained by the fact that the stored water S 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 − S).For S ≤ 0, and for = 0, i.e., P → ∞, we have = 0, B 1 = 0 and E / (P − S) = 0.When → ∞, which corresponds to P → 0, we have = 1 / H E , B 1 = 1, and E / (P − S) → 1.
Any Budyko formulation B 1 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 if S ≤ 0 then if S ≥ 0 then Figure 4 shows the ML formulation (Eq.21a, b) in the space [E p / (P − S), E / (P − S)] for ω = 1.5 and different values of H E .For H E = 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 − S) = H E E p / (P − S) when S ≤ 0, and E / (P − S) = 0 when S ≥ 0.
Hydrol ].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 H E or H P corresponds to different sets of data and different functional representations.Both approaches (H E or H P ) can be used.When storage water contributes to enhancing evaporation ( S ≤ 0), S is bounded by potential evaporation E P and consequently represents a given percentage of E P .Hence, it is more convenient to use H E = − S / E p instead of H P = − S / P , because H E lies in the range [0, 1] which is not the case for H P .Conversely, when precipitation water contributes to increase soil water storage ( S ≥ 0), S is bounded by P and represents a percentage of precipitation P .Consequently, using H P is more convenient because H P lies in the range [−1, 0].Moreover, in order to keep the parameter in the range [0, 1], H P = −H P could be preferred.
3 Comparing the new formulation with other formulae from the literature 3.1 In the standard Budyko space (E p / P , E / P ) When evapotranspiration exceeds precipitation (corresponding herein to the case S ≤ 0), Greve et al. (2016) analytically developed a Budyko-type equation where the water storage is taken into account through a parameter y 0 (0 ≤ y 0 ≤ 1) introduced into the Fu-Zhang formulation (Table 2).In the Budyko space, this equation is written (Greve et al., 2016;Eq. 9) as 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 H E and y 0 can be easily obtained.Equating Eqs. ( 14a) and ( 22) with ω = κ yields The relationship between y 0 and H E is independent from .It is shown in Fig. 5 for different values of ω.For a given value of ω, we have H E < y 0 .For ω = 1, we have H E = 0, and when ω → ∞, we have The derivative of Eq. ( 22) gives For = 0, the derivative is equal to 1, and when → ∞, the derivative tends to a value noted as m by Greve et al. (2016;Eq. 12): The value of the derivative (slope of the curve) is the same in both ML and Greve et al.'s (2016) formulations: for = 0 the derivative is equal to 1, and when → ∞ we have m = H E (assuming ω = κ).Greve et al. (2016;Sect. 4) show that y 0 is the maximum value of m reached when ω → ∞.Hence, substituting in Eq. ( 22) y 0 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 y 0 (0, 0.2, 0.4, 0.6, 0.8 and 1).The corresponding values of H E (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 S ≤ 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 y 0 .Moreover, the case of S ≥ 0 is not considered by Greve et al. (2016).2) is derived from the Turc-Mezentsev equation and written as An additional parameter t 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 − S) along the horizontal axis such as for E p / (P − S) = t , we have E / (P − S) = 0.The derivative of Eq. ( 26) when E p / (P − S) → ∞ 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 A supplementary parameter, here as µ (> −1), is added to modify the lower bound of the aridity index E P / (P − S).The parameter µ plays a similar role as t in Eq. ( 26).For µ = 0, Eq. ( 27) takes the original form of Fu-Zhang equation, (P − S) replacing P .When µ becomes positive, the lower end of the curve E / (P − S) shifts to the right.The function E / (P − S) in Eq. ( 27) is equal to zero for the particular value of E p / (P − S) = d , such as Greve et al.'s (2016) formulation can be also written in the space [E p / (P − S), E / (P − S)].Inserting Eq. ( 22) into Eq.( 20a) and expressing as a function of (Eq.18) leads to It can be mathematically shown that expressing (1 − y 0 ) in Eq. ( 29) as a function of H E 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 H E ≥ 0 are obtained with Greve et al.'s (2016) formulation.
For S ≥ 0 (corresponding to H E ≤ 0; Fig. 4), the three formulations (Chen et al.'s, 2013, Du et al.'s 2016, andML) have similar upper and lower limits.For S ≤ 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 t = d = 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 H E = + 0.25.The ML and Greve et al. (2016) formulations are exactly identical if κ = ω = 1.5 and y 0 = 0.578 calculated from Eq. ( 23) for H E = +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 − S) tends to 1 when = E p / (P − S) → ∞, while in the ML formulation with S ≤ 0 (Fig. 2a) the feasible domain is a triangle, the domain of variation of being limited by 0 and 1 / H E .
For S ≤ 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 [E p / (P − S), E / (P − S)] space, provided the shape parameters are assumed to be identical (ω = κ) (a simple relationship is established between H E and the corresponding parameter y 0 ).Despite similar upper limits, the ML and Greve et al. (2016) Du et al. (2016) The lower end of the curve E / (P − S) corresponds, respectively, to (0, 0), ( t , 0) and ( d , 0) and the upper end to (1 / H E , 1) when S ≤ 0 and (∞, 1) when S ≥ 0 for the ML formulation, (∞, 1) for the other two.Moreover, the ML formulation for S ≥ 0 is reduced to a simple relationship E / (P − S) = B 1 ( ) and is independent of H E .
It is worth noting that for S ≤ 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 P E p , and should logically tend to zero and not to t and d ; similarly, when P → 0, i.e., → ∞, it is physically logical that → E p / (− S) = 1 / H E , 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 S through a dimensionless parameter H E = − S / E p and the formulation differs according to the sign of S (Eqs.8 and 9 for S ≤ 0 and Eqs.11 and 12 for S ≥ 0).Applications can be conducted at various time steps (yearly, seasonal or monthly) both in the Turc space (P / E p , E / E p ) and in the standard Budyko space (E p / P , E / P ), with the only data required to obtain E being E p , P and S.
The new formulations are inferred from an evaluation of the feasible domain of evaporation in the Turc space, adjusted for the case where additional ( S ≤ 0) or restricted ( S ≥ 0) water is available for evaporation, and then transformed in the Budyko space.For S = 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 [E p / (P − S), E / (P − S)] space (Eq.20a, b, Fig. 4).They can also be conducted using the dimensionless parameter H P = − S / P instead of H E , which yields another form of the equations (Appendix A and the Supplement).It is shown that the ML formulation with S ≤ 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) if S ≥ 0 then 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 H P = − S / P .Figure S1 in the Supplement corresponds to Fig. 2 obtained with H E = − S / E p .

A2 General equations with restricted evaporation
We distinguish two cases: S ≤ 0 and S ≥ 0. Substituting H E by H P / in Eqs. ( 8), ( 9), ( 11) and ( 12) we obtain (if S ≤ 0 in the Turc space) (A11b) In the Supplement, Fig. S2 shows an example of the ML formulation (Eq.A11a, b) in the Budyko space obtained with the parameter H P = − S / P .It corresponds to Fig. 3 obtained with H E = − S / E p .Table S3 gives the ML formulation applied to the different Budyko curves of

Figure 1 .
Figure 1.Representation of the change in soil water storage S = (S e − S b ) / t for the two cases considered in the paper: S ≤ 0 and S ≥ 0. S b and S e are, respectively, the storage at the beginning and the end of the time period t.

Figure 3 .
Figure 3.The ML formulation in the Budyko space with the Fu-Zhang relationship Eq. (14a, b) for ω = 1.5 and for different values of H E .The bold lines indicate the upper and lower limits of the feasible domain of evaporation (shown in grey).

Figure 4 .
Figure 4.The ML formulation with the Fu-Zhang Eq. (21a, b) in the space [E p / (P − S), E / (P − S)] for ω = 1.5 and four values of H E .For H E ≥ 0, all curves have a common upper end at = 1 / H E corresponding to E / (P − S) = 1.The bold lines indicate the upper and lower limits of the feasible domain shown in grey.For H E ≤ 0, the curve is similar to the one under steady-state conditions.

Table 1 .
Different expressions for the Budyko curves under steadystate conditions.

Table 2 .
Different expressions for the Budyko curves under non-steady-state conditions.
E ): they give the ML formulation applied to the different Budyko curves of Table1in the standard Budyko space (E p / P , E / P ) and in the space [E p / (P − S), E / (P − S) . Earth Syst.Sci., 20, 4867-4879, 2016 www.hydrol-earth-syst-sci.net/20/4867/2016/ have two parameters each: one for Hydrol.Earth Syst.Sci., 20, 4867-4879, 2016 www.hydrol-earth-syst-sci.net/20/4867/2016/ the shape of the curve and another for its shift due to nonsteady conditions: ω and H E for the ML formulation (with the Fu-Zhang function), κ and y 0 for Greve et al. (2016), λ and t for Chen et al. ( formulations behave very differently from those of Chen et al. (2013) and Du et al. (2016) in the space [E p / (P − S), E / (P − S)].The ML formulation is different for S ≤ 0 and S ≥ 0, while those of Chen et al.'s (2013) and Du et al.'s (2016) do not distinguish the two cases S ≤ 0 and S ≥ 0. All the formulations have the same upper limits, but the domain of variation of differs: respectively, [0, 1 / H E ] when S ≤ 0 and [0, ∞] when S ≥ 0 for the ML formulation, [ t , ∞] for Chen et al. (2013) and [ d , ∞] for in the space [E p / (P − S), E / (P − S)].Appendix A: Scaling S by P instead of E p Appendix A presents the set of equations when scaling the change in soil water storage S by precipitation P instead of potential evaporation E p , i.e., usingH P = − S / P = H E (−1 ≤ H P ≤ ) instead of H E = − S / E p (− −1 ≤ H E ≤ 1).A1 Upper and lower limits of the Budyko frameworkIn the Turc space, the upper limits of evapotranspiration E x / E p are obtained from Eqs. (2) and (3): p P = H E = H P , A7)then in the Budyko spaceE P = B 1 ( − H P ) + H P .Replacing B 1 by Fu-Zhang's equation, in Eq. (A8) for S ≤ 0 and in Eq. (A10) for S ≥ 0, gives the following in the Budyko space: P − (1 + H P ) ω + ω 1 ω .
Table 1 with the parameter H P (Eqs.A8 and A10).It corresponds to Table S1 in the Supplement obtained with H E .The Supplement related to this article is available online at doi:10.5194/hess-20-4867-2016-supplement.