# Incorporating the Vadose Zone into the Budyko Framework

^{1}

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## Abstract

**:**

## 1. Introduction

_{0}), originally put forth by Budyko [2] (pp. 322–323). This relationship can be expressed as:

_{0}).

_{0},ΔS)

## 2. Extending the Budyko Framework

#### 2.1. The Homogeneity Postulate

_{0},ΔS) in Equation (4) is a homogeneous function of its arguments; meaning that if the independent variables are each multiplied by an arbitrary scale factor, λ, the value of f(P,ET

_{0},ΔS) is multiplied by the same scale factor, λ [4,7]:

_{0},λΔS) = λ f(P,ET

_{0},ΔS)

_{0}/P and ΔS/P, while Equation (7) constitutes the “Turc picture” of ET in the “Turc space”, i.e., ET/ET

_{0}as a function of P/ET

_{0}and ΔS/ET

_{0}[1,8]. In the present paper, only the Budyko picture will be utilized. For a detailed comparison of the Budyko and Turc pictures, see [1,8].

_{0}/P and ΔS/P are in corresponding ET states. This “law of corresponding states” [4], an example of hydrologic similarity/space-time symmetry [9], is demonstrated experimentally by the close clustering of ET data when plotted according to Equation (6). Such a test was first performed systematically by Budyko [2] (pp. 325–327) in the original framework (i.e., neglecting ΔS/P) using long-term hydrologic and climatological data for more than 1000 catchments and thereafter has been tested in a large number of careful studies [10,11,12,13,14]. The study of Ye et al. [13], which confirms the law of corresponding states in the original framework with data collected daily over half a century in more than 250 catchments located in a variety of climatic zones and physiographic regions, is exemplary.

#### 2.2. Boundary Conditions

_{0}in the energy-supply limit, Equation (8) becomes [1]:

#### 2.3. The Euler Relation

_{0},ΔS) is:

_{0}/P in the original Budyko framework.

_{0}. Equations (14) and (16) reflect the opposite tendency, i.e., the water supply is limiting and all available P is consumed by ET, which then becomes insensitive to changes in ET

_{0}. Equations (17) and (18) are unique to the extended Budyko framework. They are, however, analogous to Equations (13) and (14): ET becomes insensitive to changes in ΔS when the water supply is not limiting, akin to its insensitivity to P, whereas when the water supply is limiting, ET becomes very sensitive to changes in ΔS for the same reason that it is very sensitive to changes in P. Thus Equations (13) and (17) and Equations (14) and (18) express complementary water-balance effects on ET. The difference in sign between the right sides of Equations (14) and (18) is because ET increases as P increases, whereas ET decreases as ΔS increases.

## 3. Special Case: The Moussa-Lhomme Formulation

#### 3.1. Basic Postulates

_{0}+ ΔS) in the extended Budyko framework as ET does to ET

_{0}in the original Budyko framework, i.e., if

_{0}/P), found suitable for representing ET/P in the original framework, but with ET

_{0}/P in B(ET

_{0}/P) replaced by (ET

_{0}+ ΔS)/P in $\mathrm{B}(\frac{{\mathrm{ET}}_{0}+\mathrm{\Delta}\mathrm{S}}{\mathrm{P}})$.

_{0}/(P − ΔS) as ET/P does to ET

_{0}/P in the original Budyko framework, i.e.,

_{0}/P) in the original Budyko framework but with ET

_{0}/P in B(ET

_{0}/P) replaced by ET

_{0}/(P − ΔS) in $\mathrm{B}(\frac{{\mathrm{ET}}_{0}}{\mathrm{P}-\mathrm{\Delta}\mathrm{S}})$.

_{0}/P) found suitable for application in the original Budyko framework to represent ET/P is automatically a candidate for use in the Moussa-Lhomme (ML) extended Budyko framework. Moussa and Lhomme [1] have tabulated several such candidate functions from the literature based on both parametric and non-parametric models of ET/P.

#### 3.2. Example: Extending the Schreiber Model

_{0}/P). The Schreiber model is non-parametric, conforming to the boundary conditions in Equations (8) and (9) while interpolating smoothly between them. It is often used as a basis for comparison with parametric models [15].

_{0}/P is the well-known aridity index [3,11]. The partial derivatives in Equations (24) and (25) conform to the boundary conditions in Equations (13)–(16), with ΔS = 0. The physical basis for Equation (24) is that, for a given ET

_{0}, the sensitivity of ET to P increases with ET

_{0}, a measure of the energy that can be potentially transported to the atmosphere if P is not limiting, whereas it decreases with increasing P as the water supply for ET becomes more plentiful. The physical basis for Equation (25) is that, for a given P, the sensitivity of ET to ET

_{0}increases with P, the amount of water that potentially can be transported to the atmosphere if ET

_{0}is not limiting, whereas it decreases with increasing ET

_{0}as the energy supply for ET becomes more plentiful.

## 4. Discussion

_{0}/P and ΔS/P and thus implies hydrologic similarity/space-time symmetry [9], and Equation (12), the Euler relation, which implies that the calculation of ET/P requires only a knowledge of the two climate elasticities, ${(\frac{\partial \mathrm{ET}}{\partial \mathrm{P}})}_{{\mathrm{ET}}_{0},\mathrm{\Delta}\mathrm{S}}$ and ${(\frac{\partial \mathrm{ET}}{\partial {\mathrm{ET}}_{0}})}_{\mathrm{P},\mathrm{\Delta}\mathrm{S}}$, because the constraint imposed by the Gibbs-Duhem equation [4] makes modeling the third partial derivative in Equation (12), ${(\frac{\partial \mathrm{ET}}{\partial \mathrm{\Delta}\mathrm{S}})}_{{\mathrm{P},\mathrm{ET}}_{0}}$, redundant. Equations (22) and (23) give examples of this simplification.

_{0}+ ΔS)/P replacing ET

_{0}/P as the independent variable.

_{0}/(P − ΔS) replacing ET

_{0}/P as the independent variable.

_{0}/P), found suitable for use in the original Budyko framework, independently of whether the model contains an adjustable parameter such as w [1]. In light of this parameter independence, recent studies carried out in the original Budyko framework [3,15,23,24,25], which present statistical correlations between the Fu model parameter w and non-climatic catchment characteristics, may be premature, pending investigation of which catchment characteristics are already incorporated implicitly by considering the complete catchment water balance and including the independent variable, ΔS. Equation (28) offers a case in point, since varying ΔS/P in this equation has the same shape-changing effect on the ET/P versus ET

_{0}/P curve as does varying w in Equation (30), which neglects vadose-zone water storage.

_{0}/P is consistent with experimental data on catchment ET/P reported by others whose successful models of these data are special cases of the ML formulation [5,6,16,26]. The ML formulation is also consistent with the compilation of ET/P data for catchments in the United States reported by Sankarasubramanian and Vogel [20,21], as discussed above. Thus, further testing of the ML formulation with large data sets, particularly those that have been used in establishing statistical correlations between a model parameter such as w and catchment characteristics, is warranted and likely to be fruitful.

## Acknowledgments

## Conflicts of Interest

## Appendix A. Extending the Fu Model

_{0}, was incorporated by Greve et al. [6] into the Fu model in the original Budyko framework [19] in order to generalize it for application in an extended Budyko framework.

_{0}[19]:

_{0}, (∂ET/∂P) increases with (ET

_{0}− ET), a measure of the energy that could have been transported to the atmosphere by ET if P were not limiting, while it decreases with increasing P.

_{0}) increases with (P − ET), the volume of water that could have been transported to the atmosphere by ET if ET

_{0}were not limiting, while it decreases with increasing ET

_{0}.

_{0}and P to ET in the factors that cause the climate elasticities to increase. The expressions for the Fu model climate elasticities as modified by Greve et al. [6] through the introduction of their new parameter, y

_{0}, are:

_{0}as the maximum value of (ET − P)/ET

_{0}corresponding to “the maximum amount of additional water that contributes to ET and originates from other sources than P” [6] (p. 2199). Thus y

_{0}> 0, with its numerator representing the contribution to ET from a change in vadose-zone water storage, i.e., the numerator in y

_{0}is −ΔS and y

_{0}= −ΔS/ET

_{0}. In deriving an equation for ET/P from Equations (A1) and (A2), Greve et al. [6] followed the method in [19] to integrate them as partial differential equations. However, they assumed that y

_{0}was constant in their integration of (∂ET/∂ET

_{0}), a point which can be seen explicitly in their Equation (A13), showing y

_{0}taken outside the integral over ET

_{0}. This assumption, however, implies that ΔS is proportional to ET

_{0}, which is not consistent with ΔS being an independent variable, aside from the fact that no physical basis for ΔS being proportional to ET

_{0}is apparent. The integral over ET

_{0}performed by Greve et al. [6] should have included the term in y

_{0}under the integral sign, with y

_{0}set equal to −ΔS/ET

_{0}and ΔS, not y

_{0}, and held constant. If the integration is performed in this way, the resulting extended Fu model expression for ET/P is:

_{0}= 0, Equation (A3) reduces to the Fu model for ET/P [4,19]. Moreover, since y

_{0}> 0, the right side of Equation (A3) approaches (1 + y

_{0}ϕ) as ϕ↑∞, i.e., ET/P approaches [1 − (ΔS/P)], in agreement with Equation (11).

_{0}= 0) for the case ΔS < 0 by invoking the first hypothesis in the ML formulation, with the result:

_{0}= −ΔS/ET

_{0}.

_{0}/P from the term $\frac{{\mathrm{ET}}_{0}+\mathrm{\Delta}\mathrm{S}}{\mathrm{P}}$ inside the square brackets, is in agreement with Equation (31) as well. Thus, the model of Greve et al. [6] is a special case of the ML formulation, as was also concluded by Moussa and Lhomme [1]. However, Moussa and Lhomme [1] used the erroneous result for ET/P reported by Greve et al. [6] in making a comparison between the two models, leading them to present a spurious non-linear relationship between the parameter y

_{0}and −ΔS/ET

_{0}.

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Sposito, G. Incorporating the Vadose Zone into the Budyko Framework. *Water* **2017**, *9*, 698.
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