Crago, Richard D.

Szilagyi, Jozsef

English

DigitalCommons@University of Nebraska

University of Nebraska - LincolnDigitalCommons@University of Nebraska - LincolnPapers in Natural Resources Natural Resources, School of2019Comment on "Derivation of a sigmoid generalizedcomplementary function for evaporation withphysical constraints" by S. Han and F. TianJozsef SzilagyiUniversity of Nebraska-Lincoln, jszilagyi1@unl.eduRichard D. CragoBucknell UniversityFollow this and additional works at: https://digitalcommons.unl.edu/natrespapersPart of the Natural Resources and Conservation Commons, Natural Resources Management andPolicy Commons, and the Other Environmental Sciences CommonsThis Article is brought to you for free and open access by the Natural Resources, School of at DigitalCommons@University of Nebraska - Lincoln. Ithas been accepted for inclusion in Papers in Natural Resources by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.Szilagyi, Jozsef and Crago, Richard D., "Comment on "Derivation of a sigmoid generalized complementary function for evaporationwith physical constraints" by S. Han and F. Tian" (2019). Papers in Natural Resources. 918.https://digitalcommons.unl.edu/natrespapers/918Comment on “Derivation of a Sigmoid GeneralizedComplementary Function for Evaporation WithPhysical Constraints” by S. Han and F. TianJozsef Szilagyi1,2 and Richard D. Crago31Conservation and Survey Division, School of Natural Resources, University of Nebraska‐Lincoln, Lincoln, NE, USA,2Department of Hydraulic and Water Resources Engineering, Budapest University of Technology and Economics,Budapest, Hungary, 3Department of Civil and Environmental Engineering, Bucknell University, Lewisburg, PA, USAAbstract The sigmoid function Han and Tian derive for their E/Ep = f (Er/Ep) complementaryrelationship leads to physical contradictions, therefore cannot be accepted as an improvement of existingcomplementary relationship theory.Han and Tian attempt to derive a new complementary relationship (CR) between two nondimensionalvariables, x = Er/Ep and y = E/Ep, in the form of a preconceived sigmoid shape. Here E is the actual, whileEp (= Er + Ea) the Penman‐derived (Penman, 1948) potential evaporation rate made up of the Er energy andEa aerodynamic terms, such asEr ¼ Δ Rn−Gð ÞΔþ γ ; Ea ¼γf udvΔþ γ ; (1)where Δ is the slope of the saturation vapor pressure curve, γ the psychrometric constant, Rn the surfacenet radiation, G the ground heat flux, fu the wind function, and dv the vapor pressure deficit. FollowingCrago et al. (2016) and Szilagyi et al. (2017), Han and Tian set a constant lower limit, xmin, for x,although this lower limit clearly changes with the measurement period (Crago et al., 2016) and is defi-nitely not a constant in time. Similarly, they set another constant value, xmax, for x when the Ea termreaches its minimum. Additionally, they introduce two more parameters for regulating the shape ofthe sigmoid function.Let us note that the xmax value, when Ea is minimal under wet conditions (i.e., when water availability forevaporation is nonrestricting on a regional scale), takes up the role of the Priestley and Taylor (1972) para-meter, α, that is, Epwet = αEr = Er + Eamin = Er + Er/xmax − Er, thus xmax = α−1.The upper boundary condition (BC) proposed by Han and Tian, namely, that dy/dx = 0 at y = 1, creates ahard to interpret physical situation. Due to the flat upper part of their CR curve, E remains equal to Ep fora while as the x value starts to decrease from its xmax value. In reality, however, with the region dryingout, E can be expected to decrease from its maximum value of Ep = Epwet, attained under regionally wet con-ditions. The E = Ep condition for x close to xmax (the result of the flat upper portion of Han and Tian's curve)entails that E increases in the beginning of the drying process under a constant Er term, since the Ea term ofEp must increase to be able to move x from its maximum value of xmax. As a result, the model‐derived E notonly increases with the drying out of the environment but one also ends up with a y value above the EPT limitline (=αx), which thus loses its constraining property. Note that with xmax = α−1, point M of the EPT line inFigure 1 of Han and Tian (which corresponds to α= 1.26 in their figure) moves right horizontally, to be situ-ated above xmax, and by doing so crosses the solid red line of the sigmoid function to terminate on it (seeFigure 1 below).Let us see now how the authors arrived at their BC of (i) dy/dx = 0 at y = 1. From partial derivatives of theE/Ep = f (Er/Ep) CR equation with respect to Er and Ea, Han and Tian correctly obtained two solutions, (i)and (ii) ∂Ea/∂Er|y = 1 = α – 1, here written with the xmax = α−1 substitution. One can also obtain solution (ii)by applying the derivation with respect to Er directly on Eamin = (α – 1) Er and additionally assuming that α©2019. American Geophysical Union.All Rights Reserved.COMMENT10.1029/2018WR023502This article is a comment on Han andTian (2018), https://doi.org/10.1029/2017WR021755.Key Points:• The exact functional form of thenondimensional complementaryrelationship (CR) of evaporationremains unknown• Application of a sigmoid functionleads to physical contradictions• The slope of the CR function at theupper boundary must beconstrained by the slope of thePriestley‐Taylor limit lineCorrespondence to:J. Szilagyi,jszilagyi1@unl.eduCitation:Szilagyi, J., & Crago, R. D. (2019).Comment on “derivation of a sigmoidgeneralized complementary functionfor evaporation with physicalconstraints” by S. Han and F. Tian.Water Resources Research, 55, 868–869.https://doi.org/10.1029/2018WR023502Received 14 JUN 2018Accepted 2 JAN 2019Accepted article online 8 JAN 2019Published online 24 JAN 2019SZILAGYI AND CRAGO 868is independent of Er. However, they discard solution (ii) by saying that αmust also depend on Er. Such a direct dependence, however, has not beenshown in the literature, even though air temperature (among othervariables) may be an influencing factor on the value of α, as Han andTian correctly quote.So instead of only (i), (ii) may also be a solution to their system of equa-tions, which means, at the very least, that there exists another solutionbeside dy / dx = 0 at y = 1. In fact, due to the physical controversies thislatter solution presents, one must conclude that there is only one physi-cally interpretable solution, and that is solution (ii). Solution (ii) howeverdoes not restrict the value of dy / dx to zero at y= 1, thus a dy/dx value of αat y = 1, first proposed by Brutsaert (2015), is perfectly acceptable, as itfully avoids the physical contradictions raised by the dy/dx = 0 value ofHan and Tian.In summary it can be stated that the sigmoid function Han and Tian pro-pose for the CR relationship leads to physical contradictions, the result oftheir improper upper BC. This way one cannot consider their study animprovement upon recent CR studies by Brutsaert (2015), Crago et al.(2016), and Szilagyi et al. (2017).ReferencesBrutsaert, W. (2015). A generalized complementary principle with physical constraints for land‐surface evaporation. Water ResourcesResearch, 51, 8087–8093. https://doi.org/10.1002/2015WR017720Crago, R., Szilagyi, J., Qualls, R. J., & Huntington, J. (2016). Rescaling of the complementary relationship for land surface evaporation.Water Resources Research, 52, 8461–8471. https://doi.org/10.1002/2016WR019753Han, S., & Tian, F. (2018). Derivation of a Sigmoid Generalized Complementary Function for Evaporation With Physical Constraints.Journal of Geophysical Research: Water Resources Research, 54, 5050–5068. https://doi.org/10.1029/2017WR021755Penman, H. L. (1948). Natural evaporation from open water, bare soil, and grass. Proceedings of the Royal Society of London, A193, 120–146.Priestley, C. H. B., & Taylor, R. J. (1972). On the assessment of surface heat flux and evaporation using large‐scale parameters. MonthlyWeather Review, 100(2), 81–92. https://doi.org/10.1175/1520‐0493(1972)100<0081:OTAOSH>2.3.CO;2Szilagyi, J., Crago, R., & Qualls, R. J. (2017). A calibration‐free formulation of the complementary relationship of evaporation forcontinental‐scale hydrology. Journal of Geophysical Research: Atmospheres, 122, 264–278. https://doi.org/10.1002/2016JD025611Figure 1. Relative positions of the EPT limit line and the sigmoid functionrecommended by Han and Tian. The sigmoid function will always beabove the limit line as x nears xmax. Here an α = 1.26 value was used for theillustration, but α can take up any value typically from the [1–1.32] interval;thus, the slope of the EPT limit line changes accordingly.10.1029/2018WR023502Water Resources ResearchSZILAGYI AND CRAGO 869AcknowledgmentsThis work was supported by the BME‐Water sciences & Disaster PreventionFIKP Grant of EMMI (BME FIKP‐VÍZ).

Comment on  Derivation of a sigmoid generalized complementary function for evaporation with physical constraints  by S. Han and F. Tian

UNL | Libraries

University of Nebraska - Lincoln
DigitalCommons@University of Nebraska - Lincoln
Papers in Natural Resources Natural Resources, School of
2019
Comment on "Derivation of a sigmoid generalized
complementary function for evaporation with
physical constraints" by S. Han and F. Tian
Jozsef Szilagyi
University of Nebraska-Lincoln, jszilagyi1@unl.edu
Richard D. Crago
Bucknell University
Follow this and additional works at: https://digitalcommons.unl.edu/natrespapers
Part of the Natural Resources and Conservation Commons, Natural Resources Management and
Policy Commons, and the Other Environmental Sciences Commons
This Article is brought to you for free and open access by the Natural Resources, School of at DigitalCommons@University of Nebraska - Lincoln. It
has been accepted for inclusion in Papers in Natural Resources by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln.
Szilagyi, Jozsef and Crago, Richard D., "Comment on "Derivation of a sigmoid generalized complementary function for evaporation
with physical constraints" by S. Han and F. Tian" (2019). Papers in Natural Resources. 918.
https://digitalcommons.unl.edu/natrespapers/918
Comment on “Derivation of a Sigmoid Generalized
Complementary Function for Evaporation With
Physical Constraints” by S. Han and F. Tian
Jozsef Szilagyi1,2 and Richard D. Crago3
1Conservation and Survey Division, School of Natural Resources, University of Nebraska‐Lincoln, Lincoln, NE, USA,
2Department of Hydraulic and Water Resources Engineering, Budapest University of Technology and Economics,
Budapest, Hungary, 3Department of Civil and Environmental Engineering, Bucknell University, Lewisburg, PA, USA
Abstract The sigmoid function Han and Tian derive for their E/Ep = f (Er/Ep) complementary
relationship leads to physical contradictions, therefore cannot be accepted as an improvement of existing
complementary relationship theory.
Han and Tian attempt to derive a new complementary relationship (CR) between two nondimensional
variables, x = Er/Ep and y = E/Ep, in the form of a preconceived sigmoid shape. Here E is the actual, while
Ep (= Er + Ea) the Penman‐derived (Penman, 1948) potential evaporation rate made up of the Er energy and
Ea aerodynamic terms, such as
Er ¼ Δ Rn−Gð ÞΔþ γ ; Ea ¼
γf udv
Δþ γ ; (1)
where Δ is the slope of the saturation vapor pressure curve, γ the psychrometric constant, Rn the surface
net radiation, G the ground heat flux, fu the wind function, and dv the vapor pressure deficit. Following
Crago et al. (2016) and Szilagyi et al. (2017), Han and Tian set a constant lower limit, xmin, for x,
although this lower limit clearly changes with the measurement period (Crago et al., 2016) and is defi-
nitely not a constant in time. Similarly, they set another constant value, xmax, for x when the Ea term
reaches its minimum. Additionally, they introduce two more parameters for regulating the shape of
the sigmoid function.
Let us note that the xmax value, when Ea is minimal under wet conditions (i.e., when water availability for
evaporation is nonrestricting on a regional scale), takes up the role of the Priestley and Taylor (1972) para-
meter, α, that is, Ep
wet = αEr = Er + Ea
min = Er + Er/xmax − Er, thus xmax = α
−1.
The upper boundary condition (BC) proposed by Han and Tian, namely, that dy/dx = 0 at y = 1, creates a
hard to interpret physical situation. Due to the flat upper part of their CR curve, E remains equal to Ep for
a while as the x value starts to decrease from its xmax value. In reality, however, with the region drying
out, E can be expected to decrease from its maximum value of Ep = Ep
wet, attained under regionally wet con-
ditions. The E = Ep condition for x close to xmax (the result of the flat upper portion of Han and Tian's curve)
entails that E increases in the beginning of the drying process under a constant Er term, since the Ea term of
Ep must increase to be able to move x from its maximum value of xmax. As a result, the model‐derived E not
only increases with the drying out of the environment but one also ends up with a y value above the EPT limit
line (=αx), which thus loses its constraining property. Note that with xmax = α
−1, pointM of the EPT line in
Figure 1 of Han and Tian (which corresponds to α= 1.26 in their figure) moves right horizontally, to be situ-
ated above xmax, and by doing so crosses the solid red line of the sigmoid function to terminate on it (see
Figure 1 below).
Let us see now how the authors arrived at their BC of (i) dy/dx = 0 at y = 1. From partial derivatives of the
E/Ep = f (Er/Ep) CR equation with respect to Er and Ea, Han and Tian correctly obtained two solutions, (i)
and (ii) ∂Ea/∂Er|y = 1 = α – 1, here written with the xmax = α
−1 substitution. One can also obtain solution (ii)
by applying the derivation with respect to Er directly on Ea
min = (α – 1) Er and additionally assuming that α
©2019. American Geophysical Union.
All Rights Reserved.
COMMENT
10.1029/2018WR023502
This article is a comment on Han and
Tian (2018), https://doi.org/10.1029/
2017WR021755.
Key Points:
• The exact functional form of the
nondimensional complementary
relationship (CR) of evaporation
remains unknown
• Application of a sigmoid function
leads to physical contradictions
• The slope of the CR function at the
upper boundary must be
constrained by the slope of the
Priestley‐Taylor limit line
Correspondence to:
J. Szilagyi,
jszilagyi1@unl.edu
Citation:
Szilagyi, J., & Crago, R. D. (2019).
Comment on “derivation of a sigmoid
generalized complementary function
for evaporation with physical
constraints” by S. Han and F. Tian.
Water Resources Research, 55, 868–869.
https://doi.org/10.1029/2018WR023502
Received 14 JUN 2018
Accepted 2 JAN 2019
Accepted article online 8 JAN 2019
Published online 24 JAN 2019
SZILAGYI AND CRAGO 868
is independent of Er. However, they discard solution (ii) by saying that α
must also depend on Er. Such a direct dependence, however, has not been
shown in the literature, even though air temperature (among other
variables) may be an influencing factor on the value of α, as Han and
Tian correctly quote.
So instead of only (i), (ii) may also be a solution to their system of equa-
tions, which means, at the very least, that there exists another solution
beside dy / dx = 0 at y = 1. In fact, due to the physical controversies this
latter solution presents, one must conclude that there is only one physi-
cally interpretable solution, and that is solution (ii). Solution (ii) however
does not restrict the value of dy / dx to zero at y= 1, thus a dy/dx value of α
at y = 1, first proposed by Brutsaert (2015), is perfectly acceptable, as it
fully avoids the physical contradictions raised by the dy/dx = 0 value of
Han and Tian.
In summary it can be stated that the sigmoid function Han and Tian pro-
pose for the CR relationship leads to physical contradictions, the result of
their improper upper BC. This way one cannot consider their study an
improvement upon recent CR studies by Brutsaert (2015), Crago et al.
(2016), and Szilagyi et al. (2017).
References
Brutsaert, W. (2015). A generalized complementary principle with physical constraints for land‐surface evaporation. Water Resources
Research, 51, 8087–8093. https://doi.org/10.1002/2015WR017720
Crago, R., Szilagyi, J., Qualls, R. J., & Huntington, J. (2016). Rescaling of the complementary relationship for land surface evaporation.
Water Resources Research, 52, 8461–8471. https://doi.org/10.1002/2016WR019753
Han, S., & Tian, F. (2018). Derivation of a Sigmoid Generalized Complementary Function for Evaporation With Physical Constraints.
Journal of Geophysical Research: Water Resources Research, 54, 5050–5068. https://doi.org/10.1029/2017WR021755
Penman, H. L. (1948). Natural evaporation from open water, bare soil, and grass. Proceedings of the Royal Society of London, A193, 120–146.
Priestley, C. H. B., & Taylor, R. J. (1972). On the assessment of surface heat flux and evaporation using large‐scale parameters. Monthly
Weather Review, 100(2), 81–92. https://doi.org/10.1175/1520‐0493(1972)100<0081:OTAOSH>2.3.CO;2
Szilagyi, J., Crago, R., & Qualls, R. J. (2017). A calibration‐free formulation of the complementary relationship of evaporation for
continental‐scale hydrology. Journal of Geophysical Research: Atmospheres, 122, 264–278. https://doi.org/10.1002/2016JD025611
Figure 1. Relative positions of the EPT limit line and the sigmoid function
recommended by Han and Tian. The sigmoid function will always be
above the limit line as x nears xmax. Here an α = 1.26 value was used for the
illustration, but α can take up any value typically from the [1–1.32] interval;
thus, the slope of the EPT limit line changes accordingly.
10.1029/2018WR023502Water Resources Research
SZILAGYI AND CRAGO 869
Acknowledgments
This work was supported by the BME‐
Water sciences & Disaster Prevention
FIKP Grant of EMMI (BME FIKP‐VÍZ).


https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1927&context=natrespapers

Comment on Derivation of a sigmoid generalized complementary function for evaporation with physical constraints by S. Han and F. Tian

Abstract

Similar works

Full text

Available Versions

DigitalCommons@University of Nebraska

UNL | Libraries