A Graphical Interpretation of the Rescaled Complementary Relationship for Evapotranspiration

Abstract

Wet-surface evaporation equations related to the Penman equation can be represented graphically on vapor pressure (e) versus temperature (T) graphs (Qualls & Crago, 2020, https://doi.org/10.1029/2019wr026766). Here, actual regional evaporation is represented graphically on (e, T) graphs using the Complementary Relationship (CR) between actual and apparent potential evaporation. The CR proposed by the authors can be represented in a simple and intuitive geometric form, in which lines representing the regional latent heat flux, LE, and the wet surface (Priestley & Taylor, 1972, https://doi.org/10.1175/1520-0493(1972)100<0081:otaosh>2.3.co;2) evaporation rate, LE_PT, intersect at e = 0. This approach allows a graphical estimate of LE (or the corresponding mathematical formulation), provided available energy, wind speed, air temperature and humidity, and roughness lengths for momentum and sensible heat are available. The wet surface temperature is needed, and a calculation method for it is provided. The formulation works well using monthly data from seven sites in Australia, even when the same value of the Priestley & Taylor parameter α is used for all sites. Overall, compared to eddy covariance measurements, root mean square difference averaged 19 W m⁻²; this compares favorably with the CR formulation proposed by Brutsaert (2015, https://doi.org/10.1002/2015wr017720).

Key Points

• Latent heat flux or evaporation can be depicted on graphs of vapor pressure (e) versus temperature (T)

• The Complementary Relationship (CR) of evaporation can be expressed graphically using an e versus T graph

• The CR performs well with data from seven sites in Australia

Plain Language Summary

The Complementary Relationship (CR) between actual regional evaporation and apparent potential evaporation has received considerable support in recent years. Here, a representation of one version of the CR is developed and expressed on a single temperature/vapor pressure graph. Features such as available energy, wet surface temperature, actual surface temperature, potential temperature, and apparent potential temperature are all clearly represented graphically. The method is compared with that proposed by Brutsaert (2015, https://doi.org/10.1002/2015wr017720). Both methods performed well, but the graphical method has lower root mean squared errors.

1 Introduction

A recent summary of methods for potential and actual evaporation rates using standard meteorological data (McMahon et al., 2013) highlights the Penman (1948), Priestley and Taylor (1972), and Penman-Monteith (Allen et al., 1998) equations, noting that they are ubiquitous in the literature and highly significant for estimating potential (wet surface) or reference crop evaporation. For actual evaporation from watersheds at the monthly timescale using standard meteorological data, McMahon et al. (2013) noted that, “[t]here are a range of techniques available,” and then listed the methods developed by Brutsaert and Stricker (1979), Granger (1989), Granger and Gray (1989), Morton (1983), Morton et al. (1985), and Szilagyi and Jozsa (2008). With the possible exception of Granger and Gray (1989), all of these utilized some form of the Complementary Relationship (CR) between actual and potential evaporation, initially proposed by Bouchet (1962). This suggests that the CR had already gained the esteem of the research community. Since that time, the CR has further matured. Recent developments were sparked by the landmark paper by Brutsaert (2015), which put the CR more firmly on a physical foundation. This paper will focus on a version of the CR that grew out of the one by Brutsaert (2015).

The framework of the approach presented here comes from Qualls and Crago (2020; cf., Ma & Szilagyi, 2019). Qualls and Crago (2020) developed a graphical interpretation of the Priestley and Taylor (1972) and Penman (1948; cf., Monteith, 1981) equations and explored the theory behind a related wet surface evaporation method that is comparable to the Penman equation, but does not require the well-known assumption regarding the slope of the saturation vapor pressure curve proposed by Penman (1948). While Qualls and Crago (2020) developed concepts related to wet-surface evaporation, the present work applies the same concepts to the CR, making use of recent developments in the CR by Brutsaert (2015), R. Crago and Qualls (2018), R. Crago et al. (2016), Ma and Szilagyi (2019), and Szilagyi et al. (2016b).

In the following sections, we will review the conceptual and graphical framework for wet surface evaporation developed by Qualls and Crago (2020), then apply the same concepts to further develop and graphically describe the CR model proposed by R. Crago et al. (2016). The “generalized complementary principle” (GCP) developed by Brutsaert (2015) has been a highly cited CR formulation (e.g., Brutsaert et al., 2017, 2020; R. Crago & Qualls, 2018; R. Crago et al., 2016; Han & Tian, 2020; Szilagyi et al., 2016b; Zhang et al., 2017). To determine whether the proposed model is an improvement over the GCP, we will apply both methods to multiple years of data from seven sites in Australia and discuss the results and implications.

2 Background

The development and notation in this section closely follow that of Qualls and Crago (2020). The energy budget for land surface evaporation can be written (in W m⁻²):

(1)

where the left-hand side is the available energy, consisting of net radiation input (R_n) and heat flux into the ground, G, and the right-hand side is the sum of sensible heat flux (H) and latent heat flux LE into the lower atmosphere (Brutsaert, 2015). Latent heat flux can be written with a mass transfer equation (Brutsaert, 1982, 2005, 2015; Monteith & Unsworth, 2001; Stull, 1988) as:

(2)

where l_v is the latent heat of evaporation, e is the water vapor pressure, the subscript 0 indicates a value at the skin of the surface and subscript a indicates a value at measurement height z_a. The wind function, f(u) can be written (Brutsaert, 1982, 2005, 2015) for neutral atmospheric conditions as:

(3)

where k = 0.4 is von Karman's constant, u is the wind speed measured at height z_u, R_d is the ideal gas constant for dry air, T_a is the air temperature at z_T, d is the displacement height, z_0v is the scalar roughness length for heat and water vapor, and z₀ is the momentum roughness length. The original Penman (1948) wind function of the form f(u) = c₁ + c₂u, where c₁ and c₂ are constants, could replace Equation 3; all the theoretical developments herein would still be valid.

A heat flux equation analogous to Equation 2 can be written (Brutsaert, 2005) as:

(4)

where ρ is the density of air, c_p is the specific heat of air at constant pressure, ε = 0.622 is the ratio of the molecular weight of water to that of dry air, and T₀ and T_a are the surface skin and air temperatures, respectively.

Qualls and Crago (2020, cf., Monteith, 1981) wrote Equation 1 for a saturated surface by expressing LE with Equation 2 and H with Equation 4, resulting in:

(5)

where T_0w is the unknown surface skin temperature that would occur under the prevailing u, T_a, e_a, z₀, z_0v, d, and (R_n-G). Note that e₀ = e*(T_0w) because the air in contact with the wet surface should be saturated at the wet surface temperature T_0w. Saturated vapor pressure e* can be found from (e.g., Chow et al., 1988):

(6)

in which the temperature is in Celsius and the vapor pressure is in Pa. Provided that u, p, T_a, e_a, (R_n-G), z₀, z_0v, and d are known, Equation 5 with Equation 6 can be solved easily for T_0w using a root finder algorithm. The saturated surface latent heat flux can then be found from Equation 2 as:

(7)

where the subscript _0w indicates values at the surface with the surface assumed to be saturated.

Szilagyi and Jozsa (2008) proposed a wet surface temperature similar to T_0w, and showed that it represents the surface temperature of a small wet patch of ground within a drying region. Szilagyi and Schepers (2014) found experimentally that wet bulb temperatures (a proxy for T_0w) remain constant during regional drying, provided the available energy and wind speed remain constant. Szilagyi et al. (2016b) suggested that wet patches of any size within a drying region will all have the same value of T_0w. Conversely, this means that T_0w calculated from Equation 5 under drying actual conditions would be the same as the surface temperature if the entire region was saturated (Szilagyi et al., 2016b).

Penman (1948) developed his well-known equation for wet surface evaporation based on Equation 1-3, with the addition of the assumption that

(8)

where ∆ is the slope of the e* curve Equation 6 at T_a. With this assumption, he found:

(9)

where

(10)

γ = (c_p)p/(ε l_v) is the psychrometric constant, and p is air pressure. Note that, because of Equation 8, Equation 9 does not require a direct estimate of (T_0w). McColl (2020) uses a different assumption than Equation 8 to develop a more accurate closed-form alternative to Penman's Equation 9. In practice, such an expression could substitute for LE_0w Equation 7, but Equation 7 is retained here because theoretical developments follow from it.

Equation 4 lies, at least implicitly, behind Equations 5 and 7-10. For Equation 4 to be correct, T_a should be the potential temperature of the air at height z_T, that is, the temperature of a parcel with temperature T_a brought adiabatically down from z_T to the surface. Thus, in Equation 4, 5, and 7-10, T_a should be replaced with T_a + g∙z_T/c_p (Qualls & Crago, 2020). This correction is minimal for small values of z_T (less than a few meters), but can be significant for measurements above tall forest canopies. Neutral atmospheric stability is assumed at the monthly scale, following, for example, Brutsaert (1982, 2005, 2015) and McMahon et al. (2013).

Over a large saturated surface, Slatyer and McIlroy (1961) suggested that E_A should approach zero so that the evaporation rate from Equation 9 approaches the equilibrium evaporation rate:

(11)

Priestley and Taylor (1972) noted that even very large wet surfaces maintain evaporation rates above LE_e, and suggested that actual LE under “advection free” (more commonly called “minimal advection”) conditions is:

(12)

where α is bounded by 1 ≤ α ≤ (1 + γ/∆) based on physical constraints (Priestley & Taylor, 1972).

Equations 9 and 12 can be evaluated over actually saturated surfaces, or for unsaturated surfaces, in which case they are intended to give the apparent evaporation a wet patch would provide in a drying region (in the case of Equation 9) or the evaporation rate an entire region would have if the region was saturated (in the case of Equation 12). This is similar to the interpretation of apparent potential evaporation by Kahler and Brutsaert (2006).

Figure 1 illustrates graphically many of the concepts contained in Equations 1-12 as well as some in the following section. A brief explanation of Figure 1 follows shortly, but see Qualls and Crago (2020) for more details. The variables included in Figure 1 are further explained in Tables 1 and 2 and the paragraphs following them.

Figure 1

Graphical depiction of evaporation concepts. The graph was developed for the following values: T_a = 18°C, relative humidity = 0.2, f(u) = 2.6 × 10⁻⁷ s m⁻¹, (R_n−G) = 200 W m⁻², p = 100 kPa, and α = 1.26. The labeled points and the lines between them are given in Tables 1 and 2. Adapted from Qualls and Crago (2020).

Table 1. Points Labeled in Figure 1 (Adapted From Qualls & Crago, 2020)

Point	Description
(Ta, ea)	Measured air T and e
[Taw, e*(Taw)]	Intercept of air isoenthalp with e* curve; Taw = wet bulb air T
[T0w, e*(T0w)]	Intercept of surface isoenthalp with e* curve; T0w = wet surface T
[Ta, e*(Ta)]	Denotes e* evaluated at Ta;
(T0p, e0p)	Intersection between tangent to e* at Ta and the surface isoenthalp. Used in LEpen
(Tae, eae)	Intersection between tangent to e* at T0w and the air isoenthalp. Used in LEe
(TaPT, eaPT)	Intersection between line drawn from [T0w, e*(T0w)] at slope de/dT = s∆, where s is from Equation 13, and the air isoenthalp. Used in LEPT
(T0, e0)	An estimate of actual skin T and e. Used to find actual LE
(Tl, 0)	Zero-e intercept of line through [T0w, e*(T0w)] at slope de/dT = s∆, where s is from Equation 13. Also intercept of line through (T0, e0) and (Ta, ea)

Table 2. Lines on Figure 1 (Adapted From Qualls & Crago, 2020)

Line	Description
e*	Saturation vapor pressure curve
Air isoenthalp	Passes through (Ta, ea) with slope of –γ
Surface isoenthalp	Located ε(Rn-G)/[p cp f(u)] to the right, but parallel to, the air isoenthalp
Penman's ∆ line	Starts at [Ta, e*(Ta)] with slope tangent to e*(Ta), intercepting the surface isoenthalp at [T0p, e0p]
LE0w	Connects [T0w, e*(T0w)] to (Ta, ea), so LE0w = lv f(u)[e*(T0w)−ea]
LEpen	Connects (T0p, e0p) to (Ta, ea) so LEpen = lv f(u)[e0p−ea]
LEe	Connects [T0w, e*(T0w)] to (eae, Tae) so LEe = lv f(u)[e*(T0w)−eae]
LEPT	Connects [T0w, e*(T0w)] to (eaPT, TaPT) so LEPT = lv f(u)[e*(T0w)−eaPT]
LEX	Actual regional evaporation from the CR y = X, so LEX = lv f(u)(e0−ea)
LEPT extension	Extends the LEPT line to the zero-e intercept at Tl
LEX extension	Extends the LEX line to the zero-e intercept at Tl
LEmax (not shown)	Vertical distance from e*(T0w) to the e = 0 axis

Szilagyi and Jozsa (2008) suggested that Δ in Equation 11 should be evaluated at the air temperature that would prevail if the entire region was saturated. Since H is likely to be small in this context, one might expect this temperature to be similar to T_0w (Qualls & Crago, 2020). An alternative definition of LE_e is that it is the minimal evaporation rate possible from a saturated surface at a given available energy (Qualls & Crago, 2020).

This latter concept can be expressed graphically (Figure 1). Qualls and Crago (2020) showed that a line through (T_a, e_a) with slope de/dT = −γ is an isobaric line of constant enthalpy. That is, all points along this line represent the same energy (or enthalpy) in the forms of sensible heat (through T) and latent heat (through e). Because this constant-enthalpy line passes through (T_a, e_a), it is called the air isoenthalp. This line intersects the e* curve at the wet bulb air temperature T_aw. The surface isoenthalp is a parallel line located a distance in temperature units of ε(R_n-G)/[p∙c_p∙f(u)] to the right of the air isoenthalp (see Equation 4), passing through [T_0w, e*(T_0w)]. It represents possible surface skin temperatures and vapor pressures. Straight lines connecting the surface isoenthalp to the air isoenthalp start at points on the surface line (T₀, e₀) and end on the air line (T_a, e_a). Both points can conceivably fall anywhere along their respective isoenthalps, and any two such points automatically produce values of H = p∙c_p∙f(u)(T₀−T_a)/ε and of LE = l_v∙f(u)(e₀−e_a) that satisfy the energy budget Equation 1.

Furthermore, since the isoenthalps represent constant enthalpy and we know from Equation 5 that the line from [T_0w, e*(T_0w)] to (T_a, e_a) produces H and LE values that satisfy Equation 1, it follows that any two points on the surface and air isoenthalps will also satisfy Equation 5. In particular, the line connecting the actual (drying surface) skin temperature and vapor pressure [(T₀, e₀), both of which are typically unknown] to (T_a, e_a) gives the actual latent heat flux denoted LE_X by means of Equation 2. Note that γ depends weakly on temperature, so the isoenthalps are actually not quite straight; however, assuming they are straight is a very good assumption. In fact, when plotted at the scale of Figure 1, any deviation from a straight line is difficult to detect even with a straight edge (Qualls & Crago, 2020).

Minimum wet surface evaporation is represented by a tangent to the e* curve at T_0w, extending to its intercept with the air isoenthalp at point (T_ae, e_ae), which can be found algebraically using the equations of the two intersecting lines. This latent heat flux is identical to Equation 11 with Δ evaluated at T_0w. This is the minimum evaporation rate because any line from [T_0w, e*(T_0w)] to the air isoenthalp with a smaller evaporation than this would need to become supersaturated and cross over the saturation e* curve (e.g., Qualls & Crago, 2020). Minimal advection wet surface evaporation (Priestley & Taylor, 1972) is given by changing the slope of this line (no longer a tangent) through [T_0w, e*(T_0w)] to de/dT = sΔ (Qualls & Crago, 2020) where:

(13)

The corresponding latent and sensible heat fluxes LE_PT and H_PT are defined by Equations 2 and 4, by following the slope de/dT = sΔ from [T_0w, e*(T_0w)] to the intercept of this line with the surface isoenthalp (see Figure 1) at point (T_aPT, e_aPT) which can be found algebraically from the equations of these two lines. The formulas for LE_e and LE_PT are given in Table 2.

Figure 1 also includes a tangent line between [T_a, e*(T_a)] and (T_0p, e_0p) representing the Penman assumption Equation 8 and a line from (T_0p, e_0p) to (T_a, e_a) representing the Penman equation. Note that the Penman assumption results in a surface at (T_0p, e_0p) that is not quite saturated (see Qualls & Crago, 2020).

3 Complementary Relationship

The CR introduced by Bouchet (1962) between actual and apparent potential evaporation has a long and rapidly developing history. It is predicated on the idea that a drying region will have restricted evaporation which is reflected in overlying air that is drier and warmer than it would be if the regional surface was saturated. Thus, the demand for evaporation can be used to diagnose the rate of regional evaporation. The symmetric CR for evaporation (e.g., Brutsaert & Stricker, 1979) can be written:

(14)

where LE is the actual regional evaporation rate, LE_p is the apparent potential evaporation rate, or the latent heat flux from a small saturated patch in the region with atmospheric conditions as well as (R_n-G) identical to the actual conditions. The wet surface evaporation rate LE_w is the evaporation rate the region would have, with its given available energy (R_n-G), if (hypothetically) the region was saturated, with the atmosphere adjusted to the wet surface. More recent developments (discussed below) have largely replaced Equation 14.

In the advection-aridity approach (Brutsaert & Stricker, 1979), LE_p is taken to be LE_pen Equation 9 and LE_w is taken to be LE_PT. While these definitions are widely accepted (e.g., Brutsaert, 2015), multiple authors have noted various shortcomings with the symmetric CR Equation 14, including Brutsaert and Parlange (1998), Brutsaert et al., (2020), Han & Tian (2017, 2020), Hobbins et al. (2004), Kahler and Brutsaert (2006), Lhomme and Guillioni (2006, 2010), and Pettijohn and Salvucci (2009). In light of these shortcomings, recent developments have largely replaced Equation 14.

Specifically, in a highly influential paper, Brutsaert (2015) suggested using x = LE_w/LE_p and y = LE/LE_p to non-dimensionalize the CR. He proposed, on physical grounds, the boundary conditions y = 0 and dy/dx = 0 as x→0, along with y = 1 and dy/dx = 1 as x→1, leading to

(15)

as the lowest order polynomial that satisfies the boundary conditions, where the subscript _B indicates y as estimated from Equation 15. Equation 15, with LE_p and LE_w defined as described above for the advection-aridity approach, has been used successfully by Brutsaert et al. (2017), Liu et al. (2016), and Zhang et al. (2017).

However, Szilagyi et al. (2016a), R. Crago et al. (2016), and R. Crago and Qualls (2018) noted a difficulty with the boundary conditions as x→0. Specifically, y→0 does not imply x→0, because LE_p does not reach infinity simply because y→0. Instead, R. Crago et al. (2016) suggested that LE_p→LE_pmax, where LE_pmax can be estimated from Equation 2 [with e₀ = e*(T_0w) and e_a = 0]. In this case, the smallest x can get is x_min = LE_w/LE_max, which forms the lower limit for x, at which LE→0. R. Crago et al. (2016) “rescaled” the CR with the transformation

(16)

and suggested the CR could be written as y = X. Thus, “X” will denote a dimensionless latent heat flux estimate found using y = X with Equation 16, and the corresponding estimate of LE will be denoted LE_X. R. Crago and Qualls (2018) used Equation 16 but calculated E_pmax with Penman's Equation 9, where the drying power Equation 10 was taken to be E_A = f(u)[e*(T_dry)−0], where T_dry = T_a + e_a/γ, and ∆ was also estimated at T_dry.

4 Theoretical Development

R. Crago and Qualls (2018) suggested two primary dimensionless variables to characterize the CR. They noted that

(17)

is a representation of the energy made available when LE decreases below LE_w, and

(18)

is a representation of the energy that must be added to increase LE_p above LE_w. The denominators of both R_e and R_p are the maximum possible values of their respective numerators, both R_e and R_p reach a value of 0 when the region is saturated, and both reach a value of 1 when the regional surface is desiccated. Thus, it is reasonable to assume that R_e and R_p are the two controlling dimensionless variables in the regional drying process, so the CR can be represented as

(19)

Each term of Equations 17-19 can be written by using Equation 2. Specifically, to write LE_p, e₀ is given by e*(T_0w) and e_a is the actual vapor pressure of the air. For LE_w, e₀ is written as e*(T_0w) and e_a is e_aPT. For LE_pmax, e₀ is e*(T_0w) and e_a = e_l, where e_l is the limiting value of e that would occur if regional evaporation was zero. Substituting into Equation 19 results in

(20)

which implies

(21)

where f₁ and f₂ are functions to be determined. It is not clear how to estimate e_l, but it is presumably very small so the assumption that e_l = 0 appears reasonable. Note that the left-hand side of Equation 21 and the argument of f₂ both go to zero for a regional evaporation rate of zero, and both go to one for a saturated regional surface. These arguments suggest the simple expression:

(22)

Equation 22 could be re-written y/x = e_a/e_aPT.

Note that X in Equation 16 involves x and x_min, which in turn depend on LE_p, LE_w, and LE_pmax. Each of these can be expressed using Equation 2 with the appropriate values of e, as described below Equation 19. With these substitutions, y = X is easily shown to be identical to Equation 22. Equation 22 looks similar to the formulation of Yan and Shugart (2010), but is conceptually distinct.

Qualls and Crago (2020) discussed the extension of latent heat flux lines below and to the left of the air isoenthalp (Figure 1), and noted that this extension corresponds to hypothetical (T, e) pairs for progressively higher elevations within the constant-flux (surface) layer. With this interpretation, e_l appears to be the vapor pressure at the top of the surface layer or within the mixed layer under very dry surface conditions, such that if e₀ = e_l, regional evaporation would be zero.

An informative and intuitive geometrical interpretation of y = X or Equation 22 emerges, as shown in Figure 2, which is adapted from Figure 1 by deleting the extraneous information. It can be shown that the extensions of the LE_PT line and the LE_X line both reach a vapor pressure of e = 0 at the same temperature, T_l. This means that y = X can be expressed graphically on Figure 1 by extending the LE_PT line down to the T axis at (T_l, 0), and then drawing a line from (T_l, 0) through (T_a,e_a) to (T₀, e₀). The latent heat flux then is (l_v)f(u)[e₀−e_a]. Fluxes from this method are identical to Equation 22 or y = X with X given by Equation 16. Intuitively, looking at the LE_X extension line, an LE_X value of 0 would imply that this line is horizontal, so that (e₀−e_a) is 0, which means this line falls along the e = 0 axis of Figure 1; conversely, maximum regional evaporation would occur if the LE_X line was superimposed on top of the LE_PT line. Intermediate values of LE_X would result in the extension lines having intermediate slopes in between these two extremes.

Figure 2

Same as Figure 1, but with extraneous lines and points removed to illustrate the graphical solution to LE_X = LE_0w (e_a/e_aPT). As the region dries, (T_a, e_a) slides down and to the right along the air isoenthalp, so the slope of the dotted line decreases and so does LE_X.

The simplicity and explanatory power of the y = X assumption follows from this graphical interpretation. Given (R_n-G), f(u), T_a, and e_a, the estimate of LE based on y = X (called LE_X) is easily constructed graphically (see Figure 1):

1. The air isoenthalp passes through (T_a, e_a) at a slope of de/dT = –γ.

2. The surface isoenthalp is a distance ε(R_n−G)/[p∙c_p∙f(u)] degrees to the right of, and parallel to, the air isoenthalp.

3. The wet surface temperature T_0w is given by the intercept of the surface isoenthalp with the e* curve.

4. Starting at [T_0w, e*(T_0w)], a line having a slope de/dT = sΔ [where s is found using Equation 13 and Δ = de*/dT evaluated at T_0w] is drawn, extending all the way to e = 0 at a temperature T_l. (Note that this line intercepts the air isoenthalp at (T_aPT, e_aPT), but neither the (T, e) values at this point nor the value of LE_PT need to be known to complete the graphical construction, only the slope of this line.)

5. A second straight line, starting at the known point (T_l, 0) and passing through (T_a, e_a) is extended to the surface isoenthalp, to the point (T₀, e₀).

6. The actual latent heat flux is calculated as LE_X = l_v∙f(u)(e₀−e_a).

When z_T is high enough that T_a differs significantly from the potential temperature of the air, the graphical CR shown in Figure 2 still applies, except the point (T_a, e_a), through which the air isoenthalp passes, should be replaced with [(T_a + g∙z_T/c_p), e_a], as discussed above and by Qualls and Crago (2020).

Note that, while LE_pa and either LE_pmax or x_min play key roles in the derivation of Equation 22, none of them are found in Equation 22, and none are needed to estimate LE using Equation 22. Likewise, none of them need to be known in order to use the graphical method. In actual practice, the calculation of e_aPT in Equation 22 is comparable in difficulty to estimating x_min and X, so the advantage of Equation 22 is largely conceptual rather than computational. Also, note that LE_pmax is proportional to [e*(T_0w)−e_l], so the concept of LE_pmax is incorporated into Figure 2.

In the following section, the y = X using Equation 16 and the y_B = 2x²−x³ Equation 15 versions of the CR will be tested using data from surface flux stations in Australia.

5 Methods

5.1 Data

Data from seven FLUXNET sites in Australia were used, spanning a wide range of environmental conditions, from low crops to tall forests, from arid to humid, and from tropical to temperate climates. This is the same data set described by R. Crago and Qualls (2018), except monthly, rather than weekly, data are used here. Longer averaging times are often assumed to satisfy the assumptions behind the CR better (e.g., Szilagyi et al., 2016a, 2016b), and monthly time steps are commonly used in potential evaporation and CR studies (e.g., McMahon et al., 2013). See Han and Tian (2020) for a discussion of how averaging times affect the form of the CR.

Each site measured air temperature, vapor pressure, wind speed, net radiation, ground heat flux, and eddy covariance estimates of H, LE, and friction velocity u_*. Monthly average data were downloaded from the Fluxdata web site (http://fluxnet.fluxdata.org/). A summary of the sites is found in Table 3. FLUXNET provides data quality assessment and gap-filling by multiple methods; the method that replaced occasional missing data with ERA5 reanalysis estimates (Hersbach et al., 2020) was selected. The eddy covariance Bowen ratio was assumed to be correct, and reference (or measured) values of sensible heat flux (H_M) and latent heat flux (LE_M) were found using Equation 1 with this Bowen ratio (see R. Crago & Qualls, 2018). Site characteristics are briefly described at the FLUXNET internet site (cited above), with more details provided at the OzFlux web site (www.ozflux.org.au, last accessed 1/5/2021). The fluxes from the sites are assumed to be representative of the surrounding region.

Table 3. Description of the Sites (Adapted From R. Crago & Qualls, 2018)

Site/Fluxnet ID	Latitude (°S)	Longitude (°E)	IGBP class	z0 (m)	Tower height (m)	Months of data
Riggs Creek/AU-Rig	36.65	145.58	Grassland	0.021	2.5	33
Sturt Plain/AU-Stp	17.15	133.35	Grassland	0.05	4.8	69
Fogg Dam/(AU-Fog)	12.54	131.31	Permanent wetland	0.15	14.5	30
Ti Tree East/AU-TTE	22.29	133.64	Open Shrubland	0.28	9.8	15
Howard Springs/AU-How	12.5	131.15	Woody savanna	2	23	81
Tumbarumba/AU-Tum	35.66	148.15	Evergreen broadleaf forest	4.6	70	149
Wallaby Creek/AU-Wac	37.43	145.19	Evergreen broadleaf forest	7.4	110	21

5.2 Data Processing

At each site, months having negative net radiation, available energy, H_M, or LE_M were rejected, as were months missing any of the data listed above. The roughness length for momentum z₀, was found using the logarithmic velocity profile equation (e.g., Brutsaert, 2005) and the eddy covariance estimate of u_* for each month. For this calculation, it was assumed (prior to evaluation of the data) that d₀/z₀ = 4.8, and that the scalar roughness length is z_0v = z₀/15 (e.g., Brutsaert, 1982, 2005; 2015; Zhang et al., 2017). In this way, α remains the only tunable parameter. The log-average of the resulting monthly z₀ values was used in Equation 3. Since several of the sites had measurement heights of several tens of meters, all values of T_a were converted to potential temperatures as discussed previously.

Following R. Crago and Qualls (2018), both dimensionless (X and y_B) and dimensional (LE_X, LE_B) estimates of latent heat flux were compared with measured values (y_M, and LE_M, for dimensionless and dimensional values, respectively). Two separate methods were used to estimate LE_p, LE_w and LE_pmax. In the first method, Equation 5 was solved for T_0w, then LE_0w (estimated from Equation 7) was used for LE_p, and LE_pmax was found with Equation 2 where e₀ = e*(T_0w) and e_a = e_l = 0. Wet surface evaporation rate LE_w was found using the equation for LE_PT from Table 2; this is equivalent to Equation 12 with Equation 11, where ∆ is evaluated at T_0w. The resulting LE_p, LE_w and LE_max were used with Equations 15 and 16 to estimate latent heat fluxes. We denote these estimates of dimensionless fluxes as y_B0w and X_0w and evaluate them by comparing them to reference values y_M = LE_M/LE_0w, and we will denote the dimensional fluxes as LE_X0w and LE_B0w and compare them to reference (measured) values LE_M.

The second method used circumvents the primary disadvantage of LE_0w compared to LE_pen, that LE_0w requires a root finder to determine T_0w. Specifically, LE_p was estimated with LE_pen from Equation 9, where ∆ was evaluated at T_a. Wet surface evaporation LE_w was found using Equation 12 with Equation 11, where ∆ was estimated at T_a (rather than at T_0w as for LE_PT [see Table 2]). For LE_max, Equation 9 with Equation 10 was used, where ∆ was estimated at T_dry, E_A was taken as f(u)[e*(T_dry)-0], and T_dry = T_a + e_a/γ is the e = 0 intercept of the air isoenthalp (R. Crago & Qualls, 2018). We denote these estimates of dimensionless fluxes as X_pen and y_Bpen, and evaluate them by comparing them to reference values y_Mpen = LE_M/LE_pen, and we will denote the dimensional fluxes as LE_Xpen and LE_Bpen and compare them to reference values LE_M.

For each of these latent heat flux variables, the root mean square difference (RMSD), or the square root of the mean of the square of the difference between the estimate and the reference values, was calculated. The RMSD was calculated for a range of values of α. The calibrated value of α is that which minimizes RMSD of the chosen variable (y_B, X, LE_X, or LE_B). A different value of α was found for each of the eight estimates (X, y_B, LE_X, LE_B, each using both the LE_0w and the LE_pen method), but in each estimate, the same value of α was applied to all seven sites.

6 Results

Results are shown in Figures 3-8. In Figures 7 and 8, R is the coefficient of correlation, and “Slope” and “Intercept” are the linear regression slope and intercept: model_value = Slope * reference_value + Intercept. Notice that dimensionless y_M values using LE_0w are different than those using LE_pen, because different estimates of apparent potential evaporation are used; this is not the case for dimensional estimates.

Figure 3

Comparison between dimensionless reference values and model values. Panel “a” plots X_0w against y_M, and panel “b” plots y_B0w against y_M. Data points (color/symbol/site): (Black/x/Riggs Creek); (Green/+/Sturt Plain); (Blue/downward-pointing triangle/Fogg Dam); (Red/upward-pointing triangle/Ti Tree East); (Magenta/left-pointing triangle/Howard Springs); (Orange/circle/Tumbarumba); (Purple/square/Wallaby Creek). The black line is one-to-one.

Figure 4

Same as Figure 3, except for dimensional latent heat fluxes. Both panels have one data point outside the range of the graph: in the panel “a” at (212, 153) W m⁻²; in panel “b” at (212, 167) W m⁻².

Figure 5

Same as Figure 3 except using the LE_pen method.

Figure 6

Same as Figure 4 except using the LE_pen method. Both panels have one data point outside the range of the graph: in panel “a” at (153, 212) W m⁻²; in panel “b” at (175, 212) W m⁻².

Figure 7

Performance statistics and parameter values for various dimensionless complementary relationship methods. Slope and intercept use linear regression: model = reference * Slope + Intercept.

Figure 8

Same as Figure 7, except for dimensional complementary relationship methods. The given root mean square difference and Intercept values have been multiplied by 0.1 so lines could be plotted on a single vertical axis.

The application of the same value of α to each of the sites differs from the approach used by R. Crago and Qualls (2018), but fits with the approach of Priestley and Taylor (1972), who attempted to find a single value appropriate to a number of sites under conditions of minimal advection over a wet surface. This will be discussed more in the discussion section.

Additional calculations used α = 1.26 (the default value suggested by Priestley & Taylor, 1972) in all the models. The resulting RMSD values for X_0w and X_pen were smaller, and R values were greater, than those for y_B0w and y_Bpen with α = 1.26; in fact, they were better than the y_B0w and y_Bpen values with optimal α values (Figure 7). Likewise, RMSD and R were better for LE_X0w and LE_Xpen using α = 1.26 than for LE_B0w and LE_Bpen using the optimal values of α shown in Figure 8.

7 Discussion

7.1 Comparison of the Generalized and the Rescaled Versions of the CR

A comparison of the statistics for the models involving X with the corresponding models involving y_B in Figure 7 shows that the rescaled versions (those with X and LE_X in the column heading) perform better than the generalized form (y_B and LE_B) when a single value of α is used for all the sites. Even when the values of α that provide the optimal results for LE_B and y_B are used to estimate LE_X and X, the latter give smaller RMSD and larger R values than the former (results not shown). Additional calculations used α = 1.26 (the default value suggested by Priestley & Taylor, 1972) in all the models. The resulting RMSD values (not shown) for X_0w and X_pen were smaller, and R values were greater, than those for y_B0w and y_Bpen with α = 1.26; in fact, RMSD was lower and R greater than the y_B0w and y_Bpen values from Figure 7 using the optimal α values. Likewise, RMSD and R (not shown) were better for LE_X0w and LE_Xpen using α = 1.26 than for LE_B0w and LE_Bpen using the optimal values of α shown in Figure 8.

This strong performance by versions of the rescaled CR fits with the findings of R. Crago and Qualls (2018) using the same data set (but the current study uses monthly rather than weekly values). Those earlier results allowed for the calibration of site-specific values of α, and the improvement of the rescaled CR Equation 16 over the generalized Equation 15 was less conclusive than the present findings. Apparently, the generalized model benefitted from the use of calibrated, site-specific values of α (in R. Crago & Qualls, 2018) more than the rescaled model did, so that when the data from all sites were pooled and a single value of α was used for all sites, performance of the generalized model decreased more than that of the rescaled model.

7.2 Use of LE_pen Instead of LE_0w

Figures 7 and 8 also compare CR versions based on the LE_0w method with those based on the LE_pen method (both described above). For the generalized model Equation 15, the LE_0w method gives somewhat better results than the LE_pen method (for example, y_B0w does better than y_Bpen in terms of lower RMSD combined with higher values of R). However, for the rescaled model (versions labeled X_0W, X_pen, LE_X0w, and LE_Xpen in Figures 7 and 8), the formulations using LE_pen and LE_0w have comparable performances, although LE_Xpen has a somewhat smaller (better) RMSD than the LE_X0w version does.

On the other hand, as shown by Qualls and Crago (2020), and in the theory section of this paper, LE_0w has clear theoretical advantages over LE_pen, and it leads to fruitful and informative theoretical developments, such as the derivation of Equation 22 and the graphical interpretation of the CR shown in Figure 2. On these grounds the LE_0w method is preferable. Actually, Figures 7 and 8 suggest that the use of either of the two versions of the rescaled CR model is more important than the exact formulation of that model, and a case could be made for either the LE_pen or the LE_0w method. However, for the remainder of the Discussion section, we will focus mainly on the LE_0w method and its graphical interpretation.

7.3 Discussion of the Graphical Construction of the CR

If we assume that R_n−G and f(u) are held constant as a region dries (cf., Brutsaert, 2015), T_0w should also remain constant (Szilagyi & Schepers, 2014), so the regional drying process results in sliding of the point (T_a, e_a) down and to the right along the air isoenthalp. The value of LE_w = LE_PT should remain constant since T_0w is constant during dry-down. This means that the e = 0 intercept of the Priestley-Taylor line at (T_l, 0) should remain constant as well. Thus, the drying process is reflected by a clockwise rotation or slowly decreasing slope of the line from (T_l, 0) to (T_a, e_a) as the latter point slides down the air isoenthalp during regional drying (Figure 2). This results in a corresponding decrease in (e₀−e_a) and thus in a decrease of LE.

The role of α in the production of actual regional LE values, and how this impacts the graphical CR construction, deserves some discussion here. The value of α has been studied extensively (e.g., Brutsaert, 1982, 2005; Lhomme & Guilioni, 2010; McNaughton & Black, 1973; McNaughton & Spriggs, 1989; Priestley & Taylor, 1972). For very large wet surfaces with minimal advection, Priestley and Taylor (1972) originally estimated α = 1.26. For truly wet surfaces with minimal advection, most estimates fall in the range 1.20 ≤ α ≤ 1.30 (Brutsaert, 2005).

Bouchet's (1962) CR concept requires that a relatively large regional LE should result in relatively high moisture content in the atmospheric boundary layer (ABL). Conversely, a relatively dry ABL implies a relatively small regional LE. Having a value of α > 1 modifies this process, because α > 1 represents the import of dry air from outside the regional ABL, largely through the entrainment of free atmosphere air. This prevents the ABL from reaching full equilibrium with the moisture status of the surface skin (e.g., Lhomme & Guillioni, 2006, 2010; McNaughton & Spriggs, 1989; Raupach, 2001).

Larger values of α mean more incorporation of dry air into the ABL, such that the humidity of the ABL is lower for a given LE than it would be at equilibrium (α = 1). This examination of the physical process suggests that, for given values of T_a, (R_n−G), and f(u), values of α > 1 should correspond to greater entrainment of dry air and thus lower e_a for a given value of LE than if α was equal to 1. Increasing the value of α reduces the anticipated value of e_a. Alternatively, if all else is equal, a given (T_a, e_a) would be expected to result from a larger LE, if α is larger.

A properly formulated CR should reflect this. According to the graphical CR model illustrated in Figure 2, T_0w and α together determine the slope of the line from [T_0w, e*(T_0w)] to (T_l, 0). The point (T₀, e₀) lies at the intersection of the surface isoenthalp with the line that passes from (T_l, 0) through (T_a, e_a) (see Figure 2). With this graphical interpretation, a larger α corresponds to a steeper slope (larger de/dT) of the line through [T_0w, e*(T_0w)] and (T_aPT, e_aPT), and thus to a larger T_l. This in turn gives a steeper LE_X line and a larger value for (e₀-e_a) and a larger regional LE because of Equation 2. The dynamic illustrated in Figure 2 thus fits our understanding of how ABL dynamics impact α and regional LE.

While it is not uncommon to treat α as a local or regional variable to be determined by calibration (Brutsaert et al., 2017; R. Crago & Qualls, 2018; R. Crago et al., 2016; Zhang et al., 2017), Priestley and Taylor (1972) treated local values of α as instances of a more universal constant, arriving at the well-known value α = 1.26 through averaging the local estimates. The relatively narrow range (Brutsaert, 2005) of estimates noted above seems to support this. Conversely, Brutsaert (2005) also discusses the inherent inhomogeneity and unsteadiness of actual ABL processes, which seem likely to result in some variability in α.

It would appear that different free atmospheric conditions and ABL growth rates might result in different values of α. There is a long history of treating α as a constant with a value in the vicinity of 1.26, both in studies of wet-surface evaporation (Brutsaert, 2005; Eichinger et al., 1996; Priestley & Taylor, 1972), and in studies of the CR (e.g., Brutsaert & Stricker, 1979; Szilagyi, 2015, 2018a, 2018b; Szilagyi et al., 2016b). At the same time, other studies suggest that it is not constant (e.g., Lhomme, 1997; Pereira, 2004; Viswanadham et al., 1991) or needs to be locally calibrated (e.g., Ma, et al., 2015). McMahon et al. (2013) provide a relatively recent review of the literature.

While some of this evidence points to the advantages of local calibration of α, the need for local calibration severely limits the applicability of the CR, particularly in data-scarce regions. This consideration supports the use of a single constant value of α, provided reasonable results come from it, as they do here. Note that, although the optimal values of α found for the rescaled model in this study were slightly below the typical range (1.20 ≤ α ≤ 1.30) found by Brutsaert (2005), the present estimates are comparable to the values of 1.13 (Szilagyi et al., 2016b) and 1.15 (Szilagyi, 2018a) obtained from long-term (1979–2015) reanalysis data from the wettest of the 334 Hydrologic Unit Code level-6 watersheds in the contiguous United States; these latter estimates were independent of the CR.

Looking at the two lines passing through (T_l, 0) in Figure 1, it is clear that changing the value of α and thus of T_l will have a larger impact when LE is very close to LE_PT, than when LE is very close to zero, because the slope of the line connecting (T_l, 0) to (T₀, e₀) is relatively sensitive to T_l when the regional air is very moist, and not as sensitive when the regional air is relatively dry (far down and to the right on the air isoenthalp). In the limit as e_a→0, T_l has no impact on LE.

Note that the graphical interpretation of the CR could also incorporate a non-zero value for e_l, the minimal value of e corresponding to a regional value of LE = 0. In this case, T_l would be the intercept of the LE_PT line with a horizontal line at e = e_l. Then the line through (T_l, e_l) and (T_a, e_a) would intersect the surface isoenthalp at (T₀, e₀) and (e₀-e_a) in Equation 2 would give the regional LE.

8 Conclusions

The CR is emerging as an essential conceptual tool for researchers seeking to estimate evaporation from a region (Ma et al., 2019; McMahon et al., 2013). Beginning with the concepts of potential and apparent potential evaporation, the CR compares these two versions of wet surface evaporation and infers a regional actual evaporation rate. In this study, theoretical considerations and their graphical representation suggested by Qualls and Crago (2020) are applied to evaluate the rescaled CR (R. Crago & Qualls, 2018; R. Crago et al., 2016; Ma & Szilagyi, 2019; Ma et al., 2019; Szilagyi et al., 2016b). By writing potential and apparent potential evaporation in the form of mass transfer equations involving the vapor pressure at the surface skin and aloft in the surface layer, this study shows that the rescaled CR formulation y = X can be expressed through lines and points on a graph of vapor pressure versus temperature. A purely graphical version of the CR, along with its mathematical expression, is described. This rescaled CR is shown to provide estimates of regional dimensional and dimensionless evaporation rates better than those provided by the generalized CR Equation 15 for seven flux stations in Australia.

R. D. Crago and Qualls (2013), discussed the contributions of Wilfried Brutsaert to intuitive or conceptual models in hydrology, in particular to the CR and to the conservation of evaporative fraction during the daytime. They pointed out that intuitive concepts allow investigators to see or grasp a topic in a new and profound way, without necessarily requiring a comprehensive understanding of all the underlying physical processes. We hope the graphical approach taken here and in Qualls and Crago (2020) will help researchers gain an intuitive grasp of wet surface evaporation and of the CR, and by means of that insight to further improve our collective understanding of the evaporation process.

Acknowledgments

The authors thank the editor and anonymous reviewers for their thoughtful comments that greatly improved the manuscript. This research was supported in part by NIFA grant IDA01584.

Open Research

Data Availability Statement

The data used in this paper came from the FLUXNET project (FLUXNET, 2014) and were obtained through the Fluxdata web site (https://fluxnet.fluxdata.org/).