4.5: Reference Crop ET

Reference crop ET is defined as the ET rate from a large expanse of a uniform canopy of dense, actively growing, vegetation provided with an ample supply of soil water. The reference is a hypothetical crop (vegetation) (Allen et al., 1998). Two references are commonly used: (1) a short crop (grass clipped to maintain a height of 5 inches) and (2) a tall crop (alfalfa that is about 20 inches tall). For this text, the short reference crop is used. Other terms have been used to represent the amount of energy in the environment that is available to evaporate water. Potential ET was widely used historically to represent this energy. Currently some authors are beginning to use reference surface ET. Both terms are synonymous with reference crop ET. Reference crop ET can be predicted using a standardized equation that utilizes appropriate coefficients and standardized procedures. Numerous methods have been developed to estimate reference crop ET (ET_o). The simplest methods generally use average air temperature. The most complex methods require hourly data for solar radiation, air temperature, wind speed, and vapor pressure. There are many approaches between these extremes. The Penman-Monteith equation (Jensen and Allen, 2016) has proven to be reliable for computing reference crop ET for most locations. The Penman-Monteith equation and associated relationships for calculating coefficients were presented by Allen, et al. (1998). They utilized a short reference crop and presented procedures for either daily or hourly computations. Only the daily version of the procedure is presented here. The short reference crop and daily time steps can be used for many situations. If computations are necessary for mountainous or coastal regions, readers should refer to Allen, et al. (1998) or Jensen and Allen (2016) for appropriate methods. The Penman-Monteith equation to predict water use of the reference crop was given by Allen, et al. (1998) as:

[\dfrac{∆(R_n-G)+K_t\rho_ac_p\left(\dfrac{e_s-e_a}{r_a}\right)}{∆+\gamma(1+r_s/r_a)}\right]/\lambda \) (4.3)

where: ET_o = ET for a short reference crop,

R_n = calculated net radiation at the crop surface,

G = soil heat flux density at the soil surface,

e_s = saturation vapor pressure of air,

e_a = actual vapor pressure of the air,

∆ = slope of the saturation vapor pressure versus temperature curve,

γ = psychrometric constant,

K_t = unit conversion constant,

ρ_a = density of air,

c_p = specific heat of air,

r_a = aerodynamic resistance of water vapor movement,

r_s = bulk resistance of crop and soil surfaces, and

\(\lambda\) = heat of vaporization of water.

The amount of energy available to evaporate water determines ET rates. However, the rate of water movement from the soil and plants into the atmosphere also depends upon the resistance to the movement of water vapor within and out of the plant canopy. Wind in the atmosphere above the crop canopy causes air movement to be turbulent which results in mixing of air in the atmosphere. Thus, any water vapor that enters the atmosphere above the canopy readily mixes with air in the atmosphere and there is little resistance to water vapor movement. The flow of air within the upper portion of the crop canopy and in the air layer immediately above the crop is much less turbulent. Since there is little mixing of air from the lower portion of the plant with the air at the top of the canopy, the only way that water vapor can leave the soil plant system is due to vapor pressure gradients. Water vapor flows from areas of high vapor pressure to locations with low vapor pressure. The rate of transfer can be estimated if the difference in vapor pressure is known along with the resistance to the flow of water vapor. The aerodynamic resistance (Figure 4.8) represents the resistance to water vapor movement in the boundary layer just above the crop. Below the boundary layer the resistance to water vapor flow is controlled by soil and plant properties. For evaporation to occur water must flow through the soil pores to reach the soil/air interface. Resistance to water vapor flow also occurs within the stomata and cuticle of plant leaves. Stomatal resistance varies with the degree of water stress that plants experience, while soil resistance varies with water content. However, for reference conditions, the combined effect of these resistances can be combined into the bulk surface resistance as in Figure 4.8. While the form of the Penman-Monteith equation in Equation 4.3 is the most accurate, it requires extensive calculations. Equations associated with calculation of variables and the derivations of constants required in Equation 4.3 are very complicated and are explained in more detail by Allen et al. (1998). An American Society of Civil Engineers task force reviewed the use of the Penman-Monteith equation for 82 site-year combinations across the United States (ASCE-EWRI, 2004). They found that a simplified form of the equation provided acceptable results while simplifying calculation procedures. The reduced form of the Penman-Monteith equation for computing daily ET for a short reference crop (clipped grass approximately 5 inches tall) is given by:

\({ET}_o=\left[\dfrac{∆R_n/62.23}{∆+\gamma(1+000634U_2)} \right]+\left[\dfrac{\gamma U_2\left(\dfrac{30.18}{T+460}\right)(e_s-e_a)/25.4}{∆+\gamma(1+0.00634U_2)}\right]\) (4.4)

where: ET_o = daily reference crop ET (in/d),

R_n = net radiation (MJ/m2 /d),

U₂ = daily wind run measured 2 m above the ground (mi/d),

T = mean daily air temperature measured at height of 1.5 to 2.5 m (°F),

e_s = saturation vapor pressure of air (kPa),

e_a = actual vapor pressure of the air (kPa),

∆ = slope of the saturation vapor pressure versus temperature curve (kPa/°C), and

γ = psychrometric constant (kPa/°C).

The left bracketed term in Equation 4.4 corresponds to the radiation component of the reference ET while the right bracketed term corresponds to the aerodynamic component. This reduced form of the Penman-Monteith equation does not include the soil heat flux. The soil heat flux for daily data can generally be neglected; however, if shorter or longer time steps (i.e., hourly, or monthly) are considered, the soil heat flux should be included into the computation of ET_o as described by Jensen and Allen (2016).

The mean daily air temperature to be used in Equation 4.4 is calculated as the average of the maximum temperature for the day (T_max) and the minimum temperature for the day (T_min). The slope of the saturation vapor pressure curve (∆) depends on the mean daily air temperature (T). Values of ∆ can be determined from Table 4.1. The value of the psychrometric constant (γ) depends on the elevation of the site that the computations represent. Values for γ are given in Table 4.2. Equations for computing ∆ and γ are provided by Allen, et al. (1998). The saturation vapor pressure (e_s) for Equation 4.4 is calculated from:

\(e_s=\dfrac{e^o(T_{max})+e^o(T_{min})}{2}\) (4.5)

where e° (T_max) and e° (T_min) are the saturation vapor pressure at the maximum (T_max) and minimum (T_min) air temperatures for the day, respectively. The actual vapor pressure (e_a) is computed as the saturation vapor pressure at the dew point temperature of the air (T_dew). The dew point temperature is usually a direct input from the weather data or is derived from the weather data, then Table 4.1 can be used with T_dew. Computation of the net radiation (R_n) for estimating ET_o involves several steps. First, the amount of solar radiation that would be received on a clear day is determined as a function of the day of the year and the elevation of the site above mean sea level. Figure 4.9 can be used to determine the clear sky radiation (R_so). Then, the net outgoing long-wave radiation (R_nl) can be determined from Figure 4.10 using the maximum, minimum, and dew point temperatures along with the ratio of the measured solar radiation for the day (R_s) compared to the clear-sky radiation for that date. The net radiation is then computed from:

R_n = (1 – α) R_n – R_nl (4.6)

where α is the albedo equal to 0.23 for the short reference crop. Use of these figures and equation 4.6 will be illustrated in Example 4.1. Determination of the reference ET using Equation 4.4 involves numerous computations. A graphical procedure has been developed to accomplish the calculations. Equation 4.4 contains two bracketed sections. The left portion, within the first set of brackets, represents the reference ET that results from solar radiation. The right portion of the equation, in the second set of brackets, represents ET due to the humidification of the air. The second portion is referred to as the aerodynamic component, i.e., the ET due to advection. Figure 4.11 can be used to determine the amount of reference ET from radiation and Figure 4.12 can be used to determine the amount of reference ET from humidifying the air. Example 4.1 illustrates the procedure.

Table 4.1. Effect of air temperature on saturation vapor pressure and slope of saturation vapor pressure.

Air Temperature, (°F)	Air Temperature, (°C)	Saturation Vapor Pressure, es (kPa)^[a]	Slope of Saturation Vapor Pressure, ∆ (kPa /°C)
20	-7	0.37	0.029
25	-4	0.46	0.034
30	-1	0.56	0.041
35	2	0.69	0.049
40	4	0.84	0.059
45	7	1.02	0.070
50	10	1.23	0.082
55	13	1.48	0.097
60	16	1.77	0.113
65	18	2.11	0.132
70	21	2.50	0.154
75	24	2.96	0.178
80	27	3.50	0.206
85	29	4.11	0.237
90	32	4.81	0.272
95	35	5.62	0.311
100	38	6.55	0.354
105	41	7.60	0.403
110	43	8.79	0.457
115	46	10.14	0.518
120	49	11.67	0.584

^[a] Note a pressure of 1 kPa = 0.145 lb/in² . The atmospheric pressure at sea level averages about 101 kPa.

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Table 4.2. Value of the psychrometric constant (γ) as a function of elevation. at the location of consideration.

Elevation Above Sea Level(ft)	Psychrometric Constant, y(kPa/C)
0	0.067
500	0.066
1000	0.065
1500	0.064
2000	0.063
2500	0.062
3000	0.060
3500	0.059
4000	0.058
4500	0.057
5000	0.056
5500	0.055
6000	0.054
6500	0.053
7000	0.052

Figure 4.9. Diagram to determine clear-sky radiation (R_so) for the northern hemisphere.

Figure 4.10. Diagram to determine net outgoing long-wave radiation

Figure 4.11. Diagram to determine reference ET from radiation from Equation 4.4

Figure 4.12. Diagram for determining reference ET from aerodynamic term for Equation 4.4.

How much water might a reference crop us for a typical day in June? Data is listed below for an average day in June.

Given: Maximum air temperature = 90°F
Minimum daily air temperature = 60°F
Dew point temperature = 56°F
Solar radiation = 25 MJ/m2/d
Wind run at 2 m height = 300 mi/d
Latitude = 40°N
Elevation at site = 3,000 ft above sea level

Find: The ET_o for June 15th at the site.

Solution

1. Use the date, latitude and elevation to determine the clear-sky radiation (R_so) from Figure 4.9: On June 15th the extraterrestrial radiation is about 42 MJ/m²/d at latitude 40°N. Tracing horizontally in Figure 4.9 to an elevation of 3,000 ft gives R_so = 32 MJ/m²/d.

2. Use the maximum, minimum, and dew point temperatures with the solar radiation in Figure 4.10 to determine the net outgoing long-wave radiation:

Long-wave radiation is about 38.3 MJ/m²/d for a perfect black-body radiator.

Use dew point and solar radiation for the fraction of the emitted black-body radiation:

\(R_s/R_{so} = \dfrac{25 MJ/m^2/day}{32 MJ/m^2/day} =0.78\)

Follow along the 56°F dew point line to the ratio of 0.78 giving emitted fraction as 11.7%. Proceeding down from the upper right and horizontally from the lower right gives the net outgoing long-wave radiation (R_nl) of 4.5 MJ/m²/d.

3. Determine net radiation using Equation 4.6:

R_n = (1 – α) R_s – R_nl = (1 – 0.23) 25 – 4.5 = 14.8 MJ/m²/d

4. Use the daily wind run, elevation, average air temperature, and net radiation to determine the amount of reference ET due to radiation term: Go upward in the lower left portion of Figure 4.11 to an elevation of 3,000 ft, then right to the average temperature ([90 + 60] / 2 = 75), then upward to the net radiation of 14.8 MJ/m²/d, and finally left to the reference ET from radiation of 0.12 in/d.

5. Use Figure 4.12 for the ET from the aerodynamic term. Enter the diagram at 2 locations and determine where the lines in the middle right portion of the figure intersect. First, the vapor pressure deficit (e_s – e_a) from the lower portion of Figure 4.12 is 1.76 kPa. Going vertically from that point to the average temperature of 75°F provides a swing point on the left middle portion of the diagram. Follow that point horizontally to the right. For the second line, enter the upper left portion of Figure 4.12 with the average temperature (75°F) and go down to the elevation of 3,000 ft. Proceed right to the wind run of 300 mi/d, then down to the intersection point in the middle right portion of Figure 4.12. This gives the ET from the aerodynamic term as 0.19 in/d.

6. The total reference ET is the sum due to the radiation and aerodynamic terms:

ET_o = 0.12 + 0.19 = 0.31 in/d