W t EVAPOTRANSPIRATION

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1 Hydrologic Sciece ad Egieerig Ciil ad Eirometal Egieerig Deartmet Fort Colli, CO EVPOTNSPITION ENEGY BUDGET METHODS. Eergy budget method are baed o the direct alicatio of the eergy budget equatio, or ome aroximatio thereof. The eergy budget equatio for a cotrol olume icludig a trairig caoy ad a thi layer of urface oil ca be writte a, W t = + h L E H G L F or = L E H 3 + where L E i the latet heat flux or eergy flux ued i eaoratio; H i the eible heat flux i.e., diffuie heat exchage betwee the lad urface ad the atmohere; ad i the aailable eergy flux deity. iclude cotributio from radiatie, adectie, ad diffuie rocee, ad eergy torage chage. Oer lad urface, the aailable eergy flux deity ca be exreed a, dw = G + h LF 4 dt where i the et radiatio flux deity at the urface; h i eergy added to the ytem by adectio; dw/dt i the chage i eergy torage of the ytem; G i the et diffuie groud heat flux out of the oil layer; ad L F i eergy ue i hotoythei. The cotributio to the et radiatio flux deity at the urface are the icidet hort wae radiatio, ; the hort wae radiatio reflected back ito the atmohere, a ; the et log wae radiatio from the atmohere aborbed by the urface, ld ; ad the log wae radiatio emitted by the urface, lu. Thu, = α + 5 ld lu The eergy adected h i made u of cotributio by reciitatio, et treamflow, eeage, ad eaoratio fluxe. The aailable eergy flux deity at the urface i artitioed ito eergy for eaoratio ad eergy for eible heatig a exreed i equatio 3. I alicatio of the eergy budget method, i ofte aroximated a G, or a oly. Howeer, care mut be take to eure that uch aroximatio are alid. Jorge. amirez

2 Stadard Eergy Budget Method. mog the may alicatio of the eergy budget method, oe of the mot widely ued i the Bowe atio Method. The Bowe atio, B o, defied a the ratio of eible heat to latet heat, H B = o L E 6 ca be determied from data o ecific humidity ad temerature i the dyamic ublayer of the atmoheric boudary layer a, B o C T T T = = γ εl e e e T e 7 where C i the ecific heat of air at cotat reure; i the urface reure; L i the latet heat of eaoratio; ε i the ratio of the ga cotat for dry air, d to that of water aor,, ad umerically ε = 0.6; e ad e are the actual aor reure at eleatio z ad z withi the dyamic ublayer; T ad T are the correodig temerature. The cotat i i C γ = 8 εl ad it i kow a the ychrometric cotat. Equatio 7 aboe wa obtaied after iokig imilarity argumet for the ditributio i.e., rofile of ecific humidity, elocity, ad temerature withi the dyamic ublayer of the atmoheric boudary layer. If the Bowe ratio i kow, the the eergy budget equatio lead to the followig equatio for the eaoratio rate, E = EBB ad o adectio 9a L + B o I order to accout for the eergy adected out of the ytem by the water a it eaorate ito the atmohere, Equatio 3, 4, ad 6 lead to a modified EBB exreio a follow, E = L + B o + C wt e T b L EBB ad adectio by eaorated water oly 9b where C w i the ecific heat of water, T e i the temerature of the eaorated water, ad T b i a referece temerature. Potetial Eaoratio. Eaoratio i a ma traort roce that reult from gradiet i the ditributio of water aor ma cocetratio i.e., a gradiet of ecific

3 humidity. Potetial eaoratio refer to the rate of eaoratio from ay large uiform urface which i ufficietly moit or wet, o that air immediately i cotact with it i fully aturated, thu leadig to the larget oible local gradiet of ecific humidity, ad coequetly the maximum rate of eaoratio for the gie coditio. Some of the imlified method itroduced below are ometime ued a meaure of otetial eaoratio. Simlified Method for Wet Surface. Whe the urface i wet, it ca be aumed that the urface ecific humidity i the aturatio ecific humidity at the urface temerature. Thi allow a aroximatio, firt rooed by Pema, that elimiate the eed for meauremet at two leel i.e., leel at z ad z, a required by the tadard eergy budget method, or the o called rofile method i.e., thoe baed o imilarity rofile i the boudary layer reeted later i thi ectio. Pema equatio i, Δ E = Δ + γ e = L = e γ + E Δ + γ W + h G t L 0 where g=c /0.6L i the ychrometric cotat at urface reure, ad where D=de /dt i the loe of the aturatio water aor reure cure. Pema equatio aume that D ca be aroximated a, de T L e e T e T Δ = dt T T T E i ometime referred to a the dryig ower of the air. It i of the geeral form, E = f u e e where f i the o-called wid fuctio ad e ad e refer to aturatio water aor reure, ad actual water aor reure of the air at eleatio z, reectiely. articular form of the dryig ower of the air i the o-called aerodyamic method or Thorthwaite-Holzma equatio, E = ρk εu l z / z o e e 3 where z o i the roughe legth of the urface, ad k i o Karma cotat k = 0.4. Pema 948 uggeted the followig form for E E = f u e e = u e 4 e 3

4 where u i i m/ ad the aor reure are i hpa yieldig E i uit of mm/day. Equilibrium Eaoratio. Whe oeraig air ha bee i cotact with a wet urface oer a log fetch, it may ted to become aor aturated uder coditio of o adectio. Thu, the dryig ower of the air hould ted to aih. Therefore, i thi limit, Pema equatio yield a lower limit to the eaoratio rate from moit urface, kow a the equilibrium eaoratio rate, ad gie by, Δ E e = 5 Δ + γ L Partial Equilibrium Eaoratio. Howeer, equilibrium coditio are ecoutered oly rarely, a air i the boudary layer i cotiually reodig to large-cale weather atter which ted to maitai a aor deficit ee oer the ocea. Thu, there i alway ome degree of adectio. Prietley ad Taylor 97 took the cocet of equilibrium eaoratio a the bai for a emirical equatio giig eaoratio from a wet urface uder coditio of miimal adectio, the o-called Prietley ad Taylor equatio for artial equilibrium eaoratio, Δ E e = αeee = αe 6 Δ + γ L Prietley ad Taylor cocluded that the bet etimate of a e i a e =.6. Emirical Equatio. Jee ad Haie 963 rooed the followig emirical equatio for the etimatio of eaotrairatio for agricultural uroe. E = at + b 7 a e where a ad b are calibratio cotat, T a i the air temerature i degree Celciu, ad e i the icidet olar radiatio exreed i equialet uit of eaorated water. That i, e = 8 L Obere that thi equatio ha a form imilar to the equilibrium eaoratio equatio aboe. The liear fuctio of temerature of the equatio rooed by Jee ad Haie ca be jutified by the quai-liear deedece o temerature of the factor Δ/Δ + γ a how i the figure below. 4

5 Δ/Δ + γ I additio, the et hort wae radiatio i trogly correlated with the et radiatio, which i the mot imortat comoet of the aailable eergy,, oer daily eriod or loger. Howeer, the equatio eed to be calibrated to the ecific regio. Baed o more tha 000 meauremet of coumtie ue i the weter U.S.rereetig mea oer eriod loger tha 5 day, Jee ad Haie obtaied the followig alue for the cotat, a = 0.05 o C - ad b = 0.078, with a correlatio coefficiet r = Modified Pema Equatio for Vegetated Surface. Vegetated urface, ee whe the egetatio i well ulied with water, caot be coidered wet, excet after raifall or dew formatio. Thu, the ecific humidity at the urface of the foliage i likely to be maller tha the aturatio alue at the correodig temerature. Therefore, Pema equatio i o loger alicable. Itroducig a reitace formulatio for eaoratio uch that, E = ρ q q 9 r, where q, q are ecific humidity at eleatio z ad z, ad r, i a reitace arameter characterizig the trafer betwee oit at eleatio z ad z, a modified Pema equatio, the o called Pema-Moteith equatio, ca be deried. I order to do o, two reitace arameter mut be defied, oe characterizig the trafer betwee the aor aturated tomatal caitie ad the urface of the leaf, ad aother characterizig the aor trafer betwee the leaf urface ad the ambiet air withi the dyamic ublayer. The former reitace i kow a the tomatal reitace, r t, ad the latter a the aerodyaimc reitace, r a. The aerodyamic reitace arameter ha the followig form, r = a [l z / zo / k] / u 0 5

6 The modified Pema equatio i, E = Δ / L + ρc e e / Lr [ Δ + γ + r / r ] t a a MSS TNSFE METHODS. Baed o argumet about imilarity rofile withi the fully turbulet dyamic ublayer of the BL, the followig exreio wa obtaied for the water aor trafer a a reult of gradiet of ecific humidity i the dyamic ublayer, where u i the o-called frictio elocity, ku q q E = ρ l z / z / u = τ o / ρ 3 I the exreio for frictio elocity, t o i the urface hear tre. From imilarity argumet i the BL, the ertical rofile of horizotal elocity i the dyamic ublayer i gie by, where k i o Karma cotat, k = 0.4. u u u = l z / z 4 k Uig aalogou argumet of imilarity rofile i the dyamic ublayer, the ertical trafer of eible heat i gie by, ku C T T H = ρ 5 l z / z 6