Evapotranspiration (ET)
 All the processes by which liquid water at/near land surface becomes
atmospheric water: Evaporation + Transpiration
 Evaporation is the transfer of H2O from liquid to vapor phase. It may
occur from water bodies, soil, and plant intercepted water. It is a
diffusive process (physical process) driven by
 Saturation (vapor density) gradient ~ (rs – ra)
 Aerial resistance ~ f(wind speed, temperature)
 Energy to provide latent heat of vaporization (radiation)
 Transpiration is the evaporation occurring from plant leaves through
stomatal openings (plant mediated evaporation). It is a biophysical
process
 Controlled flow through leaf stomata (water availability?)
 Species, temperature and moisture dependent
Evapotranspiration (ET)
 E and T are combined due to difficulties in separating the two
 E from water surfaces is relatively simple process to describe. Soil
evaporation is more difficult. Transpiration from plants is even more
complex.
 Fortunately the complex process of perspiration by mammals are not
quantitatively important.
 57% of all land “P” evaporates and oceans evaporates 112% of
directly received “P”
 In arid regions ~95% of annual “P” is lost to ET. For all North
America it is ~70%.
 Water yield can be increased if ET is reduced (e.g. up to 70 cm
increase in water yield has been observed after clearcutting in some
regions)
Four Requirements for ET
Energy
Water
Vapor
Pressure
Gradient
Wind
 Two key ingredients: water and energy
 Energy is provided by sun, water typically by local precipitation
(consider irrigation).
 Thus, ET is limited to the availability of these two
Potential and Actual ET
 PET is the ET when there is ample amount of water.
 Consider the analogy of supply vs. demand:
PET  demand
AET  supply.
 Often we cannot meet the demand  PET ≥ AET.
 AET:PET is low in arid areas, whereas AET~PET in humid areas.
 Reference PET: Amount of water lost by ET by a short green crop,
completely shading the ground, of uniform height and never short of
water. Many PET formulas are based on reference ET.
 In reality, watersheds are not entirely covered by well watered short
green crops. PET for a specific crop is obtained by multiplying
reference ET by crop coefficients
http://www.fao.org/docrep/x0490e/x0490e02.jpg
Budyko Curve
 Budyko carried out empirical analysis of the climate and water
balance of a large number of catchments around the world, and
showed that they all fitted a unique curve on the AET/P vs PET/P
(Budyko diagram). The results of the simple model essentially
produces two straight lines on the diagram:
AET/P = PET/P for PET/P < 1
AET/P = 1
for PET/P > 1
 These two straight lines
happen to be upper envelopes
for the empirically obtained
relationship obtained by
Budyko for natural
catchments.
 Horizontal grey line is the water-limit, where 100% of P becomes AET
 Diagonal grey line is the energy-limit, where 100% of available
evaporative energy (i.e., PET) is converted to latent heat.
 The orange shaded area represents ε and the blue shaded area represents
the fraction of P that becomes R.
 To the left of the dashed line are energy-limited conditions (ϕ < 1) and
to the right are water-limited conditions (ϕ > 1).
Energy
 Radiation is energy that moves through space from one object, the
source, to another object where it is absorbed. Radiation sources
are generally collections of matter or devices that convert other
forms of energy into radiation. Examples are the sun and
radioactive materials.
 Solar constant: The amount of radiation reaching the outer limits
of Earth’s atmosphere at normal incidence. It is around 1,367
W/m2 (= 1.96 cal/cm2∙min) based on average earth-sun distance. It
fluctuates about 6.9% during the year.
 Only a portion of the solar energy intercepted at the top of Earth's
atmosphere passes through to the surface
 The reflectivity of the Earth or any body is referred to as its
albedo, defined as the ratio of light reflected to the light received
from a source, expressed as a number between zero (total
absorption) and one (total reflectance).
 100% = 342 W/m2 (source: NASA)
Net Radiation
Ri
aRi Re
Ri : Incoming radiation
Re : Emission by earth’s body
(1-a)Ri
a : albedo (~0.06 for deep waters, ~0.90 for fresh
snow, 0.05 for asphalt, ~0.20 for hardwood,
~0.15 for flatwood pine plantations)
 Net radiation, which is the net input at the surface is
Rn = (1-a)Ri - Re
 All bodies emit radiation depending on its temperature
Re = e∙s ∙T4
e : emissivity (=1 for blackbody, 0.97 for water surfaces)
s : Stefan –Boltzmann constant (5.67 x 10-8 W/m2K4)
T : absolute temperature in 0K (0C + 273)
Sensible and Specific Heat
 Sensible heat: It is the portion of internal energy proportional
to temperature. It is the heat you could sense by contact or
touch
 Specific heat (capacity): A measure of how a substance’s
internal energy changes with temperature. It is the amount of
heat per unit mass required to increase the temperature of a
substance by 1 0C.
dEu / m Eu / m
Cp 

dT
T
Eu: Internal energy [ML2T-2]
m: mass [M]
T: Temperature [q] - Kelvin
Example
 The heat capacity of water at 20 0C is 4.2x103 J/kgK. If we
have 0.5 kg of water and add 12 kJ of energy how much will
the water temperature raise?
 Solution: dT = 12x103/0.5/4.2x103 = 5.8 K
Therefore, temperature will rise by 5.8 0C.
NOTE: 1 cal = 4.186 J  specific heat of water = 1 cal / gr 0C
Latent Heat
 It is the portion of the internal energy that cannot be sensed or
felt. It is he amount of internal energy that is released or
absorbed during a phase change, at a constant temperature.
 Evaporation involves liquid to vapor conversion, which
requires energy added to the water
 latent heat of vaporization: lv = 2.45x106 J/kg (at 20 0C).
Compare to Cp (4.2x103 J/kgK)
 Temperature effect:
lv = 2.501x106 – 2370T J/kg (T is in 0C)
 latent heat of melting (fusion): lm = 3.34x105 J/kg (at 0 0C)
 latent heat of freezing: –lm
 latent heat of sublimation: ls = lv + lm
Energy Balance
 Rn: Net radiation flux
[E/L2T]
 H: Heat flux conducted from
hot surface to air (sensible heat
exchange)
 G: Heat flux conducted from
hot surface to soil (conduction)
 EL: Latent heat flux due to ET
= ET*rw*lv
[L/T*M/L3*E/M]
 Q: Amount of heat energy
stored per unit area
dQ
 Rn  G  H  EL
dt
 EL  Rn  G  H 
dQ
dt
dQ 

ET   Rn  G  H 
 r wlv
dt 

Example
 Calculate ET from an open water at T=30 0C if Rn=200 W/m2
assuming no sensible heat or ground heat flux.
dQ 

ET   Rn  G  H 
 r wlv
dt 

ET  Rn r wlv
200
8
ET 

8
.
2

10
m / s  7.1 mm / day
6
1000 2.50110  2370  30


Energy Conservation
 In practice we cannot
neglect H
 Neither G for small time
scales
 In figure 1st day is wet, 2nd
day is relatively dry
 Rn is almost same
 EL ~ 2H on 1st day and EL =
H on 2nd day
 If we were to plot additional
days, as soil dries H > EL
Bowen Ratio
 The ratio of sensible heat flux (H) to latent heat flux (EL) is
called Bowen ratio, i.e.
H
CB 
EL
 H  CB  EL
EL  Rn  G  H 
EL  H  EL 1  C B   1  C B r wlv ET  Rn  G 
dQ
dt
dQ
dt
Rn  G  dQ dt
ET 
r wlv 1  CB 
C p ,air K h P
 T2  T1 
 with g 
CB  g 
( psychrometric cst.)
0.622lv K w
 e2  e1 
 Kh, Kv are heat and vapor diffusivities, Kh/Kv ~ 1
 For typical values at sea level g = 0.66 mb/0C (1 mb=100 Pa)
Example
 Calculate Evaporation at Lake Hefner, OK for 7/12/1951
Data: RS,in= 30.6 MJ/m2/day, RL,atm= 34.4 MJ/m2/day, a = 0.052, e = 0.97,
Ts=26.9 0C, Ta= 27.2 0C, P=97.3 kPa, Cp,air = 1 J/gr0K, RH = 69%
Change in heat storage and heat convection to ground was negligible
 Rn = RS,n + RL,n = 30.6(1-0.052) + 0.97*34.4 – 0.97*4.9*10-9*(26.9+273)4
= 23.8 MJ/m2/day
lv = 2.501x106 – 2370T = 2.43 MJ/kg=2430 J/gr
C p ,air P  T2  T1 
1  97.3  26.9  27.2 

 
CB 

  0.0176
0.622lv  e2  e1  0.622 * 2430  3.55  2.45 
Rn  G  dQ dt
23.8
ET 

 9.97 mm / day
r wlv 1  CB 
1000 * 2.43 * 1  0.0176
If H ignored 9.79 mm/day
ET from Different Surfaces
 Open Water: Unlimited supply of water. Evaporation rate is
equal to potential rate, i.e. PET=AET
 Bare-Soil: Following infiltration due to rain, snowmelt, or
irrigation soil dries by drainage and evaporation. Evaporation
generally occurs in two stages:
(i) Atmosphere controlled stage: It is at PET and independent of soil
water content.
(ii) Soil Controlled stage: Depends on the soil water content. ET is
less than free-water rate.
The transition from stage (i) to (ii) is quite abrupt and can be
visually detected as an increase in brightness (albedo) of soil.
Transpiration
 Transpiration: It involves absorption of soil water by plant
roots, translocation in liquid form through the vascular system
of the roots, stem, and branches to the leaves and eventually to
the walls of tiny stomatal cavities where evaporation takes
place. Water vapor moves into ambient air through stomata.
 Plants live by absorbing CO2 from air. CO2 enter plant in
dissolved form. Stomatal cavities provide this opportunity.
 Air in stomatal cavities is saturated at the temperature of the
leaf, and water movement is due to vapor pressure difference.
 Major difference between transpiration and open-water
evaporation is that plants can exert some physiological control
over the size of the stomatal openings, thus ease of vapor
movement, by the action of guard cells.
More on Transpiration
 Major factors effecting the opening and closing of guard cells are:
1. Light: Most plants open stomata during day and close at night
2. Humidity: Stomatal openings tend to decrease as humidity decreases
below its saturation value
3. Water content of the leaf cells: If daytime water contents become too
low, stomata tend to close
 Transpiration is a physical, not a metabolic process:
 When water exits through stomata potential energy decreases.
 Water moves up through vascular system creating water content gradient
between the root and the soil
 Water moves from soil to roots resulting in reduction in soil water
content in adjacent soil
 Water in soil moves towards roots
Transpiration Dominates the Evaporation
Process
Trees have:
•Large surface area
•More turbulent air flow
•Conduits to deeper moisture sources
T/ET
Hardwood ~80%
White Pine~60%
Flatwoods ~75%
The driving force
of transpiration is
the difference in
water vapor
concentration, or
vapor pressure
difference,
between the
internal spaces in
the leaf and the
atmosphere
around the leaf
Transpiration
 The physics of evaporation from stomata are the same
as for open water. The only difference is the
conductance term.
 Conductance is a two step process
 stomata to leaf surface
 leaf surface to atmosphere
Transpiration
How Does Water Get to the Leaf?
Water is PULLED, not pumped.
Water within the whole plant
forms a continuous network of
liquid columns from the film of
water around soil particles to
absorbing surfaces of roots to
the evaporating surfaces of
leaves.
It is hydraulically connected.
Even a perfect vacuum can only pump
water to a maximum of a little over 30
feet. At this point the weight of the
water inside a tube exerts a pressure
equal to the weight of the atmosphere
pushing down
So why doesn’t the continuous
column of water in trees taller
than 34 feet collapse under its
own weight? And how does
water move UP a tall tree against
the forces of gravity?
> 100 meters
Water is held “up” by the surface tension of tiny menisci (“menisci” is the plural
of meniscus) that form in the microfibrils of cell walls, and the adhesion of the
water molecules to the cellulose in the microfibrils
cell wall microfibrils of carrot
Cohesion-Tension Theory:
(Böhm, 1893; Dixon and Joly, 1894)
The cohesive forces
between water molecules
keep the water column
intact unless a threshold of
tension is exceeded
(embolism). When a water
molecule evaporates from
the leaf, it creates tension
that “pulls” on the entire
column of water, down to
the soil.
?
ET = Rain * 0.80
ET = Rain * 0.95
1,000 mm * 0.80 = 800 mm
G = 200 mm
1,000 mm * 0.95 = 950 mm
Assume Q & ΔS = 0
G = P - ET
G = 50 mm
4x more groundwater recharge from open stands than from
highly stocked plantations.
NRCS is currently paying for growing more open stands, mainly for wildlife.
Penman Method
 Considers both aerodynamic and energy effects
ET 

g
Er 
Ea
 g
 g
Er: Evaporation rate due to net radiation (L/T)
Ea: Evaporation rate due to aerodynamic effects (L/T)
g: psychometric constant
: gradient of the saturated vapor pressure at air temperature
 Evaporation rate is weighted sum of a rate due to radiation and
a rate due to mass transfer
 It is the most widely used and generally most accurate method
for free water evaporation  Became the standard
Penman Method
ET 
4098es

237.3  T 2
Er 
Rn
r wlv

g
Er 
Ea
 g
 g
T in 0C
Ea  Bes  e
g
CpP
0.622lv
e  Rh es
es  611e
17.27T
237.3T
0.622 r a k 2u z
B
2
Pr w ln  z z0 
k=0.4 (von Karman Constant)
ra and rw = density of air (1.19 kg/m3 at 25 0C) and water
P = Air pressure
z0 = surface roughness height
z= elevation where wind velocity is measured
B: Water vapor transfer coefficient
Example
 Rn= 200 W/m2, Ta= 25 0C, Rh= 40%, uz= 3 m/s at z=2 m, z0= 0.3 mm
Er 
200
 8.22 108 m / s  7.1 mm / day
6
(2.5 10  2370  25)  997
0.622 1.19  0.42  3
11
B

4
.
54

10
m / Pa  s
3
2
101.3 10  997  [ln( 2000 / 0.3)]
es  611 exp(
17.27  25
)  3167 Pa
237.3  25
e  0.4  3167  1267 Pa
Ea  4.54 10 11  (3167  1267)  8.62 10 8 m / s  7.45 mm / day
1005 101.3 103
0
g

67
.
1
Pa
/
C
3
0.622  244110
E

4098  3167
0

188
.
7
Pa
/
C
2
(237.3  25)
188.7  7.1  67.1 7.45
 7.2 mm / day
188.7  67.1
Penman - Monteith
 It is the modification of the Penman method by including a
factor for soil surface and/or stomatal diffusion resistance to
account for resistance to vapor-flux through stomata and
unsaturated soil
ET 
 Er  g Ea
  g (1  rc / ra )
1 ln 2 z z0 
ra: diffusion resistance factor of air, ra  2
k
uz
rL
rc: canopy resistance, rc 
0.5LAI
LAI: Leaf area index (maximums: conifer=10, broadleaf=6, grass=4)
rL: Effective stomatal resistance of a single leaf (use minimum
rL for PET: conifers = 1/1.6 sec/mm, broadleaf = 1/2.5
sec/mm, grass = 1/5.0 sec/mm)
Simpler Methods:
 Hamon model: Daily PE (mm/day) is given by
PET  29.8d
es
T  273
where d is day length in hr, and es is saturation vapor pressure in kPa at the
mean daily temperature T (0C).
 Priestly-Taylor: Over large areas the second term in Penman’s equation is
approximately 30% of the first term
PET  a

Er
 g
where a = 1.3
 Hargreaves: Daily ET in mm/day is given by
PET  0.0023Re T  17.8 Tmax  Tmin
PET and Re have same units:
(Divide PET by lrw for mm/day)
where Re is extraterrestrial radiation (function of day & latitude), T= 0C
Actual ET: AET
 When water supply for vaporization is deficient or soil
moisture is below the field capacity, then vaporization cannot
proceed at the potential level. Thus AET is fraction of PET:
AET = x ∙(PET)
where x is affected not only by soil moisture content but also
by climate and species.
 Calculation of AET from PET is complicated
 In open water AET=PET
 For soil AET, soil moisture needs to be followed closely
 Plant AET not only depends on soil moisture but also
photosynthetic activities
Forest Impact
 Transpiration and evaporation in a forested watershed occur at
different rates
 Evaporation of canopy-intercepted water often exceeds open water
evaporation
 Lu et al (2003) developed the following regression model for AET
from 39 experimental watersheds in the Southeastern U.S.
AET
P
z
f
%F
: long-term annual AET (mm/yr)
: Watershed mean precipitation (mm/yr)
: mean watershed elevation (m)
: latitude at the watershed outlet (degrees)
: watershed percent forest cover
 10% increase in forest cover  20 mm/yr increase in AET
Measurement of ET
 Evaporation pans: Measures PET.
 NWS uses “Class A Pan” with h=10” and D=48”. Filled with 8” water;
when level drops 7” pan is refilled. Evaporation pans overestimate PET
by 20-40%. Pan coefficients are used to adjust PET (~ 0.7).
 Farmers typically cut a 55-gal barrel in half and bury one-half, leaving
only 2” to 3” of circular rim above ground. They keep the water level at
ground level, replacing loss each day. Knowing their soils and crops
they maintain a soil water budget
 Lysimeters: Used to measure AET from a specific
plant. It is difficult to extrapolate to larger areas.
 Two types: weighing lysimeters sits on a scale and
change in weight is measured. Non-weighing
lysimeters are based on soil water budget.
Eddy Flux
Towers