Saturday, January 3, 2015

Potential Evapotranspiration

Potential Evapotranspiration (pET) is the keystone of many papers about evaporation and ecohydrology. It is usually believed that it can be established as “a reference condition” (not to be confused with a reference evapotranspiration - rET) that can subsequently be used to determine or frame the actual evapotranspiration (aET) by introducing “resistances” that limit the potential fluxes. 

However, as Brutsaert  says, the concept is a slippery one, and actually as reported by Mac Mahon et al. “there have been many definitions and redefinitions of the term potential evaporation or evapotranspiration“. Granger [1989a, 1989b] seems to have reported at least five of these definitions “but considered only three to be useful”. 
  • the first is defined as the evaporation rate that would occur from a saturated surface with a constant energy supply to the surface 
  • the second is defined as the evaporation rate that would occur from a saturated surface with constant energy supply to, and constant atmospheric conditions over, the surface 
  • the third is defined as the evaporation rate that would occur from a saturated surface with constant atmospheric conditions and constant surface temperature 
Dingman’s (1992, Sect. 7.7.1) definition of potential evapotranspiration says that it “. . . is the rate at which evapotranspiration would occur from a large area completely and uniformly covered with growing vegetation which has access to an unlimited supply of soil water, and without advection or heating effects.” ^1

My definition of pET would be probably simpler than Dingman's one. Consider  a surface under  stationary (i.e. Fixed) meterological/thermodynamical conditions, including wind speed, air humidity, atmospheric and air temperature, and the roughness parameters. I would intend as pET the evapo(transpi)ration happening from the surface as if water supply would be unrestricted and unlimited, like if the surface were a bulk water reservoir, even if the roughness conditions would not be realistic for a water body.

To make the latter statement,  I have formulas in my mind.  Do not blame me if I do not consider as reasonable those empirical definitions of pET which, derived from observation in natural landscapes,  are based on variable thermodynamics, forcings, and turbulence. pET definition requires those conditions fixed, and, while they could be in principle determined experimentally in laboratories' cases, not in the field,  keeping all the fluxes  controlled I have information of only one successful experiment of this type, performed  by Dani Or at ETH.

Back to my equations, the starting point can be the Dalton equation (Deq, I do not claim that it is true, just saying that it is usually believed so, and I am starting from this assumption, but see, for instance here to start thinking to something different):

In Dalton’s equation, pET is when resistance is just the aerodynamic one. So in Deq, pET is pretty well defined.  
In Penman-Monteith (PM) approach, Dalton equation is simplified (for a derivation, see, for instance Entekhabi, 1997 (pg. 4-25), assuming no storage of energy, and no net advection:

So also in PM (which we can consider an approximation of Deq), pET  has a clear interpretation, which is, the ET you obtain by putting any resistance, rg (either representing soil or vegetation, indeed), to zero. Excluding the resistance (and, in case, considerations about roughness) there is no way to distinguish, either in Deq and PM a soil surface from a vegetated surface from a water body. Therefore, my definition of pET is, according to these formulas, universal. 

Unfortunately this is not the same in Priestley-Taylor (PT), which can be seen as a further simplification of PM formula, where the resistance terms simply are not present:

where everything is lumped in the alpha coefficient. What people did in this case, I think, to assess alpha in pET (say alpha_pt) was to measure ET in the field's conditions supposed to be those in which pET realizes, a modus operandi on which, I already said I am skeptic.  ( I confess I did not went into the detail of that literature, see for instance the overview by Cristea et al, 2012 or also Flint and Childs, 1991 and Eichinger et al, 1996). Once your get alpha_pt you can estimate pET. remarkably.  if we are interested in aET, the passage through pET could be not necessary, if we would have a method to directly determine the (mean) PT alpha that correspond to aET directly.

Another step to do is to remind which is the spatial and temporal validity of the above Deq, PM and PT equations. In principle, Deq should be valid over a short period of time (for which statistical turbulence characteristics can be considered assessed), so should be PM and PT. However, in literature daily, weekly and monthly abuse of them is often made without a real theoretical treatment. Especially PM (in its FAO version, Allen et al., 1998) and PT are, in fact, used as formula to regress against data, and various values of the parameter are presented as reliable. But, as we know, the more we go far away from the consistent physical origin of  equations, the more uncertainty we have in our forecasting. 

To be continued ;-). To see some shy hint to what happens when one average in space, see this previous post. To see some more though about Evapotranspiration, here is the whole collection of posts.

^1 - (All of these definitions were grabbed from the Mac Mahon’s paper, that is a source also of reference historical papers - see below)

P.S. - Concerning FAO guideline 56 (see below), there "potential" refers to the well-watered status of the grass but does not mean that surface resistance becomes zero. With the FAO definition of a reference grass surface in terms of height, LAI, stomatal resistance etc., one arrives at almost exactly 70 s/m which is also the recommended value for direct input (see e.g. p. 23/24 of FAO56).  I find it ambiguous, anyway.


Allen, R. G., Pereira, L. S., Raes, D., and Smith, M.: Crop evapo- transpiration Guidelines for computing crop water requirements, FAO Irrigation and Drainage Paper 56, Food and Agriculture Organization of the United Nations, 1998.

Cristea, N.C.; Kampf, S. K. ; and Burges, Stephen J., F.ASCE, Revised Coefficients for Priestley-
Taylor and Makkink-Hansen Equations for Estimating Daily Reference Evapotranspiration, Journal of Hydrologic Engineering, Vol. 18, No. 10, October 1, 2013. ISSN 1084-0699/2013/10- 1289-1300

Dingman, S. L.: Physical Hydrology, Prentice Hall, Upper Savage, New Jersey, 1992.

Eichinger, W.E.; Parlange, M.B.; and Stricker, H., On the concept of equilibrium evaporation and the value of Priestley-Taylor coefficient, Water Resour. Res., vol 32, No. 1, 161-164, 1996

Entekabhi, D, Land surface Processes: basic tools and concepts, p. 4-25, in Marani, M and Rigon R, Hydrometeorology and climatology, Istituto Veneto di Scienze, Lettere ed Arti, Ve, 1997

Flint, A.L.; and Childs, S.W.; Use of Priestley-Taylor evaporation equation for soil water limited conditions in a small forest clearcut, Agricultural and Forest Meteorology, 56, 247-260, 1991

Granger, R. J. and Gray, D. M.: Evaporation from natural nonsaturated surfaces, J. Hydrol., 111, 21–29, 1989.

Granger, R. J.: A complementary relationship approach for evaporation from nonsaturated surfaces, J. Hydrol., 111, 31–38, 1989b.

McMahon,T.A.; Peel, M. C.; Lowe, L.; Srikanthan, R. and T. R. McVicar, Estimating actual, potential, reference crop and pan evaporation using standard meteorological data: a pragmatic synthesis, Hydrol. Earth Syst. Sci., 17, 1331–1363, 2013 doi:10.5194/hess-17-1331-2013

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