Matching the Budyko functions with the complementary evaporation relationship : consequences for the drying power of the air and the Priestley – Taylor coefficient

The Budyko functions B1(8p) are dimensionless relationships relating the ratio E/P (actual evaporation over precipitation) to the aridity index 8p = Ep/P (potential evaporation over precipitation). They are valid at catchment scale with Ep generally defined by Penman’s equation. The complementary evaporation (CE) relationship stipulates that a decreasing actual evaporation enhances potential evaporation through the drying power of the air which becomes higher. The Turc–Mezentsev function with its shape parameter λ, chosen as example among various Budyko functions, is matched with the CE relationship, implemented through a generalised form of the advection–aridity model. First, we show that there is a functional dependence between the Budyko curve and the drying power of the air. Then, we examine the case where potential evaporation is calculated by means of a Priestley–Taylor type equation (E0) with a varying coefficient α0. Matching the CE relationship with the Budyko function leads to a new transcendental form of the Budyko function B1(80) linking E/P to 80 = E0/P . For the two functions B1(8p) and B1(80) to be equivalent, the Priestley–Taylor coefficient α0 should have a specified value as a function of the Turc–Mezentsev shape parameter and the aridity index. This functional relationship is specified and analysed.

and soil water storage (Li et al., 2013;Yang et al., 2007). The most representative functions E/P = B(Φ p ) are shown in Table   1 (see Lebecherel et al. (2013) for an historical overview) and one of them (Turc-Mezentsev) is represented in Fig. 1 for different values of the shape parameter. Steady-state conditions are assumed, considering that all the water consumed by evaporation E comes from the precipitation P and that the change in catchment water storage is nil: P-E = Q with Q the total runoff. All the Turc-Budyko functions should necessarily verify the following conditions: (i) E = 0 if P = 0, (ii) E ⩽ P 5 (water limit), (iii) E⩽ E p (energy limit), (iv) E → E p if P→ +∞. These conditions define a physical domain where the Turc-Budyko curves are constrained (Fig. 1). It is interesting to note also that any Turc-Budyko function B 1 relating E/P to Φ p can be transformed into a corresponding function B 2 relating E/E p to Φ p -1 = P/E p (Zhang et al., 2004;Yang et al., 2008). Indeed (1) 10 Potential evaporation, which establishes an upper limit to the evaporation process in a given environment, is generally given by a Penman-type equation (Lhomme, 1997a). It is the sum of two terms: a first term depending on the radiation load R n and a second term involving the drying power of the ambient atmosphere E a = + . (2) In Eq.
(2) γ is the psychrometric constant and ∆ the slope of the saturated vapour pressure curve at air temperature. E a 15 represents the capacity of the ambient air to extract water from the surface. It is an increasing function of the vapour pressure deficit of the air D a and of wind speed u through a wind function f(u): E a = f(u) D a . Contrary to precipitation, potential evaporation E p is not a forcing variable independent of the surface. E p is in fact coupled to E by means of a functional relationship known as the complementary evaporation relationship (Bouchet, 1963), which stipulates that potential evaporation increases when actual evaporation decreases. This complementary behaviour is made through the drying power 20 of the air E a : a decreasing actual evaporation makes the ambient air drier, which enhances E a and thus potential evaporation.
Eq. (2) takes into account this complementary behaviour through the drying power E a , which adjusts itself to the conditions generated by the rate of actual evaporation. It is also the case, for instance, when E p is calculated as a function of pan evaporation.
However, in most of Turc-Budyko functions encountered in the literature, E p is not accurately defined. Choudhury (1999, p. 25 100) noted that "varied methods were used to calculate E p , and these methods can give substantially different results". Many formulae, in fact, can be used to calculate the potential rate of evaporation, each one involving different weather variables and yielding different values. Some formulae are based upon temperature alone, others on temperature and radiation (Carmona et al., 2016). In the present study we examine the case where E p is estimated via a Priestley-Taylor type equation (Priestley and Taylor, 1978) with a variable coefficient α 0 : 30 = . (3) Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2016-220, 2016 (Shuttleworth, 2012), which can be seen as a direct consequence of the complementary evaporation relationship. Lhomme (1997b) made a thorough examination of the coefficient α 0 by means of a convective boundary layer model.
In the present paper, the behaviour of the drying power of the air E a will be examined, together with its physical boundaries, 5 in relation to the actual rate of evaporation predicted by the Turc-Budyko functions. It will be also shown that the coefficient α 0 has a functional relationship with the shape parameter of the Turc-Budyko curve and the aridity index. The standpoint used in this study differs from various previous attempts undertaken in the literature to examine from different perspectives the links between Bouchet and Turc-Budyko relationships, investigating their apparent contradictory behaviour. For example, Zhang et al. (2004) established a parallel between the assumptions underlying Fu's equation and the 10 complementary relationship. In a study by Yang et al. (2006) concerning numerous catchments in China, the consistency between Bouchet, Penman and Turc-Budyko hypotheses was theoretically and empirically explained. Lintner et al. (2015) examined the Budyko and complementary relationships using an idealized prototype representing the physics of large-scale land-atmosphere coupling in order to evaluate the anthropogenic influences. Zhou et al. (2015) developed a complementary relationship for partial elasticities to generate Turc-Budyko functions, their relationship fundamentally differing from 15 Bouchet's one. Carmona et al. (2016) proposed a power law to overcome a physical inconsistency of the Budyko curve in humid environments, this new scaling approach implicitly incorporating the complementary evaporation relationship.
The paper is organized as follows. First, the basic equations used in the development are detailed: the choice of a particular Turc-Budyko function is discussed and the complementary evaporation relationship, implemented through the Advection-Aridity model (Brutsaert and Stricker, 1979) is presented. Second, the feasible domain of the drying power of the air E a is 20 examined, together with the correspondence between E a and actual evaporation in dimensionless form. Third, the functional relationship linking the Priestley-Taylor coefficient α 0 to the shape parameter of the Turc-Budyko function and the aridity index is inferred. In the following development, "Turc-Budyko" will be abbreviated in TB and "complementary evaporation" in CE.

Basic equations 25
Among the TB functions given in Table 1, one particular form is retained in our study: the one initially obtained by Turc (1954) and Mezentsev (1955) through empirical considerations and then analytically derived by Yang et al. (2008) through the resolution of a Pfaffian differential equation with particular boundary conditions. Three reasons guided this choice: (i) the function is one of the most commonly used; (ii) it involves a model parameter λ which allows it to evolve within the Turc-Budyko framework; (iii) it has a notable simple mathematical property expressed as: F(1/x) = F(x)/x. This last property 30 means that the same mathematical expression is valid for B 1 and B 2 (Eq. 1). The so-called Turc-Mezentsev function is expressed as: Hydrol. Earth Syst. Sci. Discuss., doi: 10.5194/hess-2016-220, 2016 Manuscript under review for journal Hydrol. Earth Syst. Sci. Published: 23 May 2016 c Author(s) 2016. CC-BY 3.0 License.
It is written here with an exponent noted λ instead of the n generally used (Yang et al., 2009). The slope of the curve for Φ p = 0 is 1. When the model parameter λ increases from 0 to +∞, the curves grow from the x-axis (zero evaporation) to an upper limit (water and energy limits), as shown in Fig. 1. In other words, when λ increases, actual evaporation gets closer to its maximum rate and when Φ p tends to infinite E/P tends to 1. The intrinsic property of Eq. (4) allows it to be transformed Fu (1981) and Zhang et al. (2004) derived a very similar equation with a shape parameter ω (see Table 1) and Yang et al.
(2008) established a simple linear relationship between the two parameters (ω = λ + 0.72). In the rest of the paper, the development and calculations are made with the Turc-Mezentsev formulation. However, similar (but less straightforward) 10 results can be obtained with the Fu-Zhang formulation (see the supplementary material S4).
The complementary evaporation (CE) relationship expresses that actual evaporation E and potential evaporation E p are related in a complementary way following E w is the wet environment evaporation, which occurs when E = E p and b is a proportionality coefficient (Han et al., 2012). 15 Various forms of the CE relationship exist in the literature (Xu et al., 2005). In our analysis, it is interpreted in the widely accepted framework of the Advection-Aridity model (Brutsaert and Stricker, 1979), where b = 1, potential evaporation E p is calculated using Penman's equation (Eq. 2) and E w is expressed by the Priestley-Taylor equation where the coefficient α w has an estimated and fixed value of 1.26. E w only depends on net radiation and air temperature 20 through ∆. As already said in the introduction, the complementarity between E and E p is essentially made through the drying power of the air E a : a decrease in regional actual evaporation, consecutive to a decrease in water availability, generates a drier air, which enhances E a and thus E p . The fact that E 0 (Eq. 3), as a substitute for E p , should also verify the CE relationship implies that: α w ⩽ α 0 ⩽ 2α w .

Feasible domain of the drying power of the air and correspondence with the evaporation rate 25
As a consequence of the CE relationship, the drying power of the air E a is linked to the evaporation rate. Its feasible domain is examined hereafter by determining its bounding frontiers and its behaviour is assessed as a function of the evaporation rate. Inverting Eq. (2) and replacing its radiative term by E w (Eq. 7) yields to Taking into account the CE relationship (Eq. 6 with b=1) and scaling by E p leads to Inserting Eq. (5) into Eq. (9) gives This means that the ratio E a /E p can be also expressed and drawn as a function of Φ p -1 like the TB functions. Given that there 5 is a water limit expressed by 0 < E < P and an energy limit expressed by 0 < E < E p , the function E a /E p = D(Φ p -1 ) should meet the following three conditions: (i) E > 0 implies that E a < E a,x given by: (ii) E < P implies that E a > E a,n1 given by: 10 .,9 = )1 + * 1 − -, )1 + 8 *# .
With E p as scaling parameter, the feasible domain of E a /E p in the dimensionless space (Φ p -1 = P/E p , E a /E p ) is shown in Fig.   2c: when evaporation is nil, E a = E a,x is maximum (upper boundary in Fig. 2c); when evaporation is maximal, E a is minimal 15 (lower boundary in Fig. 2c). The maximum dimensionless difference D* between the upper boundary (E a,x /E p ) and the lower boundary is obtained by subtracting Eq. (13) from Eq. (11): There is a correspondence between the TB curves E/P = B 1 (Φ p ) and E/E P =B 2 (Φ p -1 ) drawn into Figs. 2a, b and the one of E a /E p = D(Φ p -1 ) drawn in Fig. 2c. Figs. 2a, b, c show this correspondence for a particular case defined by λ = 1 and T = 15°C 20 (∆ = 110 Pa °C -1 ). When the TB curves reach their upper limit, i.e. in very evaporative environments, the corresponding curve E a /E p reaches its lower limit. Conversely, when the TB curves reach their lower limit, i.e. the x-axis (no-evaporative environment), the corresponding E a /E p curve reaches its upper limit.
It is interesting to note that the parameter λ of the Turc-Mezentsev function has a clear graphical expression. Denoting by d* the maximum difference between the Turc-Mezentsev curve and its upper limit (Fig. 2a), this difference (0 < d* <1) 25 obviously occurring for Φ p = P/E p = 1, we have from Eq. (4) .
This simple relationship shows that the dimensionless differences d* and δ* vary simultaneously in the same direction with a proportionality coefficient equal to D*, whose value is close to 1. It is a direct consequence of the CE relationship. When d* decreases, i.e. the dimensionless evaporation rate (E/P or E/E p ) increases, δ* decreases, i.e. the drying power of the air E a decreases: for a constant wind speed, the air becomes wetter.
In the next section, another consequence of the CE relationship will be examined in relation to the value of the  Taylor coefficient and its dependence on the rate of actual evaporation.
For a given value of the exponent λ, a fixed value of α 0 and with α w = 1.26, the relationship between E/P and Φ 0 (or between E/E 0 and Φ 0 -1 ) can be obtained by using numerical methods to resolve Eqs. (20) and (21). Similar calculations, more or less complicated, could be made with any Turc-Budyko function. These results show that a Turc-Mezentsev curve (or any TB 5 curve) generates a different curve when potential evaporation is given by E 0 instead of E p . This new curve is represented in Fig. 3 by comparison with the original one for two values of the shape parameter λ (0.5 and 2) assuming α 0 = α w = 1.26. The new curve has a form similar to the original one, with the same limits at 0 and +∞, but it is higher or lower depending on the value of α 0 . It is worthwhile noting also that B 2 ' is different from B 1 ', contrary to B 2 (Eq. 5) which is identical to B 1 (Eq. 4).
Nevertheless the two curves are very close, as shown in Fig. 4, and it is easy to verify they have the same value for Φ 0 = Φ 0 -10 1 =1. We The same relationship (Eq. 22) is obtained by matching B' 2 with B 2 . It is worthwhile noting that when α 0 is expressed by Eq.
(22) and Φ 0 tends to zero (or Φ 0 -1 tends to infinite), α w /α 0 in Eqs. (20) and (21) and For every value of λ and Φ, a unique value of α 0 can be calculated by means of Eq. (22), α w being fixed. In this equation α 0 = f(λ, Φ), Φ represents climate aridity and λ catchments characteristics in relation to its ability to evaporate (the greater λ, the higher its evaporation capability). The Priestley-Taylor coefficient α 0 appears to be an increasing function of Φ and a 30 decreasing function of λ. Fig. 5a shows the relationship between α 0 and λ for different values of Φ. α 0 tends to 2α w when λ Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2016-220, 2016 Manuscript under review for journal Hydrol. Earth Syst. Sci. Published: 23 May 2016 c Author(s) 2016. CC-BY 3.0 License. tends to zero (non-evaporative catchment) whatever the value of Φ. When λ tends to infinity (i.e. very evaporating catchment), the limit of α 0 depends on the value of Φ. For Φ ⩽ 1 the limit is α w and for Φ > 1 the limit is the branch of the hyperbole 2α w Φ/(1+Φ). Fig. 5b shows the relationship between α 0 and Φ for different values of λ. When Φ tends to +∞ (very arid catchment), the coefficient α 0 tends to 2α w . When Φ tends to 0 (very humid catchment), α 0 tends to α w . These results illustrate the simple functional relationship existing between the Priestley-Taylor coefficient, the TB shape parameter 5 and the aridity index. Similar results are obtained when the Fu-Zhang formulation is used, as detailed in the supplementary material S4.

Summary and conclusion
The TB curves have two different and equivalent dimensionless expressions: B 1 where E/P is a function of the aridity index specified value as a function of α w , λ and Φ 0 = Φ p so that the two curves B 1 and B 1 ' be equivalent. This means that the coefficient α 0 (α w ⩽ α 0 ⩽ 2α w ) is intrinsically linked to the shape parameter λ of the Turc-Mezentsev function and to the aridity index.    Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2016-220, 2016 Manuscript under review for journal Hydrol. Earth Syst. Sci. Published: 23 May 2016 c Author(s) 2016. CC-BY 3.0 License.