Department of Biological and Agricultural Engineering, University of Idaho, Moscow, Idaho, USA

Corresponding author: R. J. Qualls, Department of Biological and Agricultural Engineering, University of Idaho, Moscow, ID 83843-0904, USA. (rqualls@uidaho.edu)

[1] The Evaporative Fraction (EF) and the Complementary Relationship (CR), both extensively explored by Wilfried Brutsaert during his productive career, have elucidated the conceptual understanding of evapotranspiration within hydrological science, despite a lack of rigorous proof of validity of either concept. We briefly review Brutsaert's role in the history of these concepts and discuss their appeal and interrelationship.

[2] Part of the legacy of Wilfried Brutsaert's hydrological research is the value he has placed on using intuitive concepts to understand hydrologic processes. This letter discusses two of the concepts he has employed in his evaporation research: the conservation of flux ratios during the daytime, and the CR between regional and potential evaporation. While neither has been rigorously proven, both have been surprisingly fruitful in promoting our understanding of evaporation processes, are widely used in research, and have been incorporated into operational models and data products (representative references will be given in the course of this letter).

[3] In this letter, we will: (1) Briefly review Brutsaert's role in the history of the conservation of flux ratios and the CR; (2) Discuss what about these concepts accounts for their appeal and fruitfulness; and (3) Discuss their interrelationship.

2. History

2.1. Conservation of Flux Ratios

[4] The evaporative fraction EF = LE/(H + LE) is the ratio of latent heat flux LE to available energy H + LE (=R_{n} − G), where H is sensible heat flux, R_{n} is net radiation, and G is ground heat flux. The value of EF indicates how favorable conditions are for conversion of available energy into latent rather than sensible heat fluxes [e.g., Brutsaert, 2005]. The concept of the “conservation,” “self-preservation,” or “self-similarity” (all terms used by Brutsaert himself) of EF during a single daytime is usually credited to Shuttleworth et al. [1989], using data from the FIFE experiment in Kansas in 1987. They observed that EF remained nearly constant on fair-weather days. They built on earlier work by Jackson et al. [1983] who took the ratio LE/R_{s}, where R_{s} is incoming shortwave radiation, as constant during the daytime [see Brutsaert, 2005]. Sugita and Brutsaert [1991] and Nichols and Cuenca [1993] also used constant EF to extrapolate from a single LE measurement to the total daytime evaporation. Brutsaert and Sugita [1992] extended the analysis to other “flux ratios” which they defined as the ratio of LE to “any other flux component,” finding that good estimates of daytime evaporation could be obtained assuming the self-preservation of EF, LE/R_{n}, and LE/R_{s}. Brutsaert and Chen [1992] plotted EF during the daytime for a succession of days during a drying period of FIFE 1987. They observed conservation of daytime EF throughout the drying period, with lower values on successive days.

[5] Brutsaert also explored reasons for the self-preservation. Brutsaert and Sugita [1992] noted that clouds disrupt the diurnal progress of EF. Crago and Brutsaert [1996] showed, by propagating uncertainties through the equations, that uncertainty in daytime evaporation estimated by assuming constant EF, is generally smaller than with constant Bowen ratio H/LE.

[6]Crago [1996] searched systematically for a physical reason for the self-preservation of EF. Other studies looked to convective boundary layer models to analyze flux ratio self-preservation [e.g., Lhomme and Elguero, 1999; Peters-Lidard and Davis, 2000; Raupach, 2000; Gentine et al., 2010, 2011a, 2011b]. Gentine et al. [2010, 2011a, 2011b] also considered temporal patterns of variability of the constituents of flux ratios. Van Niel et al. [2012] developed a general model to evaluate sources of error and to minimize bias within conservation of flux ratios [see also Van Niel et al., 2011]. Thus, there is a considerable body of work providing a theoretical understanding of the preservation of flux ratios. The consensus is that some of the flux ratios, including EF, tend to have only small variability during the daytime under conditions of relatively large, stable net radiation, and steady, low-to-moderate horizontal advection [cf. Brutsaert, 2005, p. 138].

2.2. Complementary Relationship

[7] The CR between actual regional-scale evaporation E and apparent potential evaporation E_{p} was first proposed by Bouchet [1963] as cited by Brutsaert [2005]. Apparent potential evaporation is that which would occur under current conditions if the surface was brought to saturation. If the regional surface was truly wet, E and E_{p} would be equal, and Brutsaert [2005] denotes this evaporation rate E_{p0}:

E=Ep=Ep0(1)

[8] As the surface dries, E decreases below E_{p0} by an amount ΔH, which is primarily seen as an increase in sensible heat flux:

Ep0âˆ’E=Î”H(2)

[9] The CR is based on the idea that E_{p} is increased by a fraction of ΔH above E_{p0}:

Epâˆ’Ep0=ÎµÎ”H(3)

[10] Combining these two equations gives [e.g., Kahler and Brutsaert, 2006]:

E=(1+1/Îµ)Ep0â€“Ep/Îµ(4)

[11] Most researchers until relatively recently have assumed the “effectiveness parameter,” ε [Brutsaert, 2005], is given by ε = 1. The idea behind equation (4) is that air over a region absent significant external advection is in equilibrium with the land surface, so that dryness of the air implies regional evaporation rates are low. Despite considerable work (some of which is described below), the processes behind the CR and its time scales are not fully understood.

[12]Brutsaert and Stricker [1979] developed the advection-aridity (A-A) equation representing E_{p0} with the Priestley-Taylor equation, E_{p} with the Penman equation, and ε = 1. Many researchers have used the CR [Morton, 1983; Qualls and Gultekin, 1997; Hobbins et al., 2001; Crago and Crowley, 2005; Crago et al., 2005, 2010; Szilagyi and Jozsa, 2009; Huntington et al., 2011], others have attempted to develop rigorous derivations of the CR [e.g., Granger and Gray, 1989; Szilagyi, 2001; Szilagyi and Jozsa, 2008], and still others have used models or data to test the CR assumptions [e.g., McNaughton and Spriggs, 1989; Lhomme and Guilioni, 2010; Pettijohn and Salvucci, 2009]. Evaporation from a pan is a 3-D process [Pettijohn and Salvucci, 2009], while Penman's equation is a (1)-D process. They note this increases pan evaporation above what would be predicted by Penman's equation, and this difference may play a role in the varying values of ε in the literature, that is, in the varying degrees of asymmetry in the CR.

[13] While most applications of the CR have been at daily and longer time steps [e.g., Morton, 1983; Brutsaert and Stricker, 1979], it is also possible to apply it at subdiurnal time steps. This generally requires accounting for atmospheric stability through Monin-Obukhov Similarity, as explained by Brutsaert [1982] and first applied by Parlange and Katul [1992]. The ground heat flux may also become important at time scales shorter than 24 h.

[14]Brutsaert and Parlange [1998] proposed that the CR could help explain the so-called evaporation paradox, in which pan evaporation was found to be decreasing globally over time while global warming suggested evaporation should be increasing. They suggested that pan evaporation could play the role of E_{p} in the CR, so that decreases in pan evaporation could imply an increase in actual evaporation, which would resolve the paradox. For further information, see Roderick and Farquhar [2002], Hobbins et al. [2004], Brutsaert [2006], Roderick et al. [2009a, 2009b], and Van Heerwaarden et al. [2010].

[15] The CR and the self-preservation of flux ratios have been incorporated into widely used models and data products. For example, Nishida et al. [2003] developed a remote sensing-based model that uses the CR and the self-preservation of daytime EF. This has gone into the MOD-16 Evapotranspiration data product (http://modis.gsfc.nasa.gov/data/dataprod/pdf/MOD_16.pdf). Self-preservation of flux ratios have been incorporated into the data assimilation model of Caparrini et al. [2004 a, 2004b] and the METRIC model of Allen et al. [2007].

3. Discussion

3.1. What Makes These Successful Concepts?

[16] Both concepts have a strong empirical basis—substantial evidence supports using these concepts under a range of “typical” conditions. Both are “simple and plausible” [Brutsaert, 2005], and intuitive: They have helped many researchers develop a mental framework of the evaporation process. In addition, they provide a powerful way to leverage sparse data: Flux ratio self-preservation allows a single-time-of-day remotely sensed measurement of LE to be extrapolated to a daily value; the CR allows actual regional evaporation to be estimated based solely on common climatological measurements. Finally, they help to make sense of otherwise confusing results, such as decreasing pan evaporation in a warming climate.

3.2. How Are They Related Concepts?

[17]Brutsaert [2005] prefers to classify models, methods, or theories according to spatial and temporal scales, rather than as empirical, physically based, or conceptual. The flux ratio self-preservation concept takes measurements of LE at small time scales and proposes a framework for how they represent LE over the entire day. The CR uses measurements at small spatial scales, for example at an evaporation pan, to infer what is happening over a larger homogeneous region. In fact, some derivations of the CR make use of a hypothetical small wet surface within a larger (homogeneous) drying region [e.g., Morton, 1983; Granger and Gray, 1989; Szilagyi, 2001; Szilagyi and Jozsa, 2008] which evaporates at the rate E_{p}, variously defined. In both cases, “point” measurements are extrapolated to larger temporal or spatial scales.

[18] At a deeper level, the atmospheric boundary layer (ABL) ties both concepts together. The diurnal evolution of the ABL integrates processes that occur over the entire day and over regional spatial scales; the state of the ABL at a particular time and place results from processes that occurred hours to days earlier, several tens of km upwind. Thus, temporal and spatial scales are closely related, since processes that occur far upwind affect the ABL much later in time. This close linkage is perhaps one reason why convective boundary layer models have proved useful for evaluating both concepts [e.g., Kim and Entekhabi, 1997; Lhomme and Elguero, 1999; Lhomme and Guilioni, 2010; McNaughton and Spriggs, 1989; Peters-Lidard and Davis, 2000].

[19] As a dimensionless aid to understanding the relationship between conservation of EF and the CR, Figure 1 shows EF plotted as a function of surface resistance (r_{s}/r_{a}) and drying power of the air (E_{A}/Q_{n}). In the figure r_{s} and r_{a} are surface and aerodynamic resistances to water vapor transfer, E_{A} = f(u)(e_{a}* − e_{a}) is the drying power of the air, f(u) is a function of the wind speed, found using logarithmic wind speed and vapor pressure profiles (with no stability correction), e_{a}^{*} and e_{a} are the saturated and actual vapor pressure at the air temperature, and Q_{n} is the available energy in the same units as E_{A}. In Figure 1, air temperature is 20°C, wind speed is 2 m s^{−1}, and together with humidity all are measured at 2 m. The aerodynamic roughness length is 0.02 m. Figure 1 shows the EF solution space over a wide range of the dimensionless variables using the Penman-Monteith equation [e.g., Brutsaert, 1982]. Ranges of EF are shown as bands of color on the EF surface; transitions between bands represent lines of constant EF.

[20] Figure 1 includes a curve, lying on the EF surface, corresponding to Brutsaert and Stricker's [1979] A-A model, with α = 1.26. Vertical lines drop down from this curve to the (r_{s}/r_{a}, E_{A}/Q_{n}) plane. The A-A curve was constructed by setting EF calculated by the Penman-Monteith equation equal to EF given by the A-A equation. For specific values of r_{s}/r_{a} this was solved for E_{A}/Q_{n}, and EF plotted in terms of its (r_{s}/r_{a}, E_{A}/Q_{n}) coordinates.

[21] Figure 2 projects the A-A curve of Figure 1 onto the (E_{A}/Q_{n}, r_{s}/r_{a}) plane (middle solid curve) and provides two other solid curves for air temperatures of 15 and 25°C. Values of EF are shown to the right of each data point. The dotted lines follow contours of constant EF (see the figure caption for details). In the following, we assume that the A-A lines in Figures 1 and 2 correctly represent regional evaporation with minimal advection. However, the concepts discussed would apply to most versions of the CR.

[22] In Figure 1, conservation of EF at constant temperature requires movement along the surface parallel to the contours of constant EF (the colored bands on the surface), while the CR (A-A) concept requires movement only along the A-A curve. Satisfaction of both at the same time requires r_{s}/r_{a} and E_{A}/Q_{n} to remain constant during the day, which seems unlikely. However, daytime conservation of EF and adherence to the CR are both possible with air temperature changes. This can happen if diurnal movement follows a constant-EF curve (such as one of the two dashed lines) as the temperature changes. Thus, EF can remain constant during a period of warming (cooling) while A-A is also maintained, provided both r_{s}/r_{a} and E_{A}/Q_{n} increase (decrease) as indicated by the dashed lines. In fact, there are often observed increases of r_{s} from stomatal constriction due to increased stress including warmer temperatures and increased vapor deficit [e.g., Campbell and Norman, 1998]. Thus, although there are clearly circumstances (such as strong advection) which cause deviations from the patterns described above, Figure 2 shows how daytime conservation of EF can be consistent with the CR.

[23] Over a series of days during a drying cycle, the change in EF specified by the A-A curves in Figures 1 and 2 will likely describe the observed time rate of decrease of actual mean daily EF (e.g., as illustrated in Brutsaert [2005, Figure 4.12]) as the ABL temporally adjusts to the drying surface conditions. We think this is a key link between these two seemingly different concepts.

4. Conclusions

[24] As Wilfried Brutsaert has demonstrated throughout his career, an intuitive understanding of hydrological processes is indispensable in hydrological research. Intuitive thinking frames questions, prioritizes options, and evaluates results. As Brutsaert [2005] noted, all models of hydrological processes are intuitive or conceptual at some level. Consider Darcy's law: It is considered a “first principle” in many branches of hydrology, but it is really a macroscale simplification of the fluid mechanics of flow through an irregular matrix of soil particles, and fluid mechanics is a macroscale result of the laws governing molecular motion.

[25] The CR and self-preservation of flux ratios are generally not considered “first principles” in hydrology. Clearly, these approaches are imperfect and conditions for their validity may rarely be perfectly met. Yet they have aided our intuition about the evapotranspiration process for decades. This letter explored their history, reasons for their usefulness, and their shared dependence on multiscale ABL processes. We suspect that they will play a role in hydrology for years to come.