Theoretical analysis of the effects of irrigation rate and paddy water depth on water and leaf temperatures in a paddy field continuously irrigated with running water

Agricultural Water Management 198 (2018) 10–18

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Theoretical analysis of the effects of irrigation rate and paddy water depth on water and leaf temperatures in a paddy field continuously irrigated with running water Kazuhiro Nishida ∗ , Shuichiro Yoshida, Sho Shiozawa Department of Biological and Environmental Engineering, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-8657, Japan

a r t i c l e

i n f o

Article history: Received 28 November 2016 Received in revised form 2 November 2017 Accepted 24 November 2017 Keywords: Paddy field irrigation Thermal environment in paddy fields Heat balance equation Analytical solution Heat damage of rice

a b s t r a c t Water management techniques such as continuous irrigation with running water (CIRW) have been used by Japanese farmers to control the thermal environment for rice cropping. In this study, to optimize this method to control the thermal environment in paddy fields, theoretical equations for predicting water and vegetation temperatures in a paddy field were obtained. First, the equations to calculate water and vegetation temperatures in a paddy field were obtained by solving the heat balance equations of the paddy water and vegetation, taking into account the effect of horizontal heat convection driven by irrigation. The equations were validated by comparisons with observed water temperatures in a conventional paddy field under CIRW. The calculated changes in water temperatures over time and distance showed good agreement with observed values, with a root mean square error of 0.39 ◦ C. This result indicated that the equations satisfactorily expressed the features of paddy water temperature under CIRW. Next, these equations were used to determine the effects of irrigation rate, paddy water depth and wind speed on water and rice plant temperatures. The following results were obtained. 1) The area cooled by CIRW was positively related to irrigation rate and negatively related to water depth. 2) Low water depths were preferable for application of CIRW during the nighttime. 3) Slower wind speeds and latent heat flux from vegetation strengthened the effects of water management on vegetation temperature. The proposed equations and analyses represent the common features of the effect of irrigation on water and vegetation temperatures in a paddy field, and provide quantitative information about the effects of water management techniques on the thermal environment for rice cropping. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Water and air temperatures in paddy fields substantially affect rice growth, yield, and quality. In Japan, recent high air temperatures during the rice grain ripening period have decreased the quantity and quality of the rice crop. Consequently, high temperatures have become a major concern for rice cultivation (Morita 2008; Terashima et al., 2001). High temperatures increase the percentages of immature and cracked rice grains (Nagata et al., 2005; Tashiro and Wardlaw 1991; Terashima et al., 2001). These types of damage reduce the overall grade of rice in quality surveys (Morita 2008), and thus reduce the income of farmers. It is feared that future global warming will expand the area where high-temperature damage occurs (Morita 2008). Accordingly, countermeasures are required to prevent high-temperature damage to rice plants.

∗ Corresponding author. E-mail address: [email protected] (K. Nishida). https://doi.org/10.1016/j.agwat.2017.11.021 0378-3774/© 2017 Elsevier B.V. All rights reserved.

Several water management techniques in which water temperature is controlled to ensure appropriate growth conditions have been proposed as countermeasures to prevent heat damage to rice crops in Japan. These methods include continuous irrigation with running water (CIRW) and controlling water depth (e.g. saturated irrigation, deep water irrigation) (Tomosho and Yamashita 2009). In particular, CIRW using cool water has attracted considerable attention because of its potential to reduce or eliminate high-temperature damage to rice. CIRW irrigates water at temperatures lower than those of the surrounding air and paddy water, and simultaneously drains the paddy water to allow the cool water to replace the paddy water and cool the rice plant body. The Ministry of Agriculture, Forestry and Fisheries of Japan (2008, 2011) reported that use of CIRW as a countermeasure against high summer temperatures reduced rice damage in Japan. Similarly, several studies (Arai and Ito, 2001; Arai et al., 2006; Nagahata et al., 2005; Nagata et al., 2005; Miyasaka et al., 2011; Wada et al., 2013) have investigated the relationship between CIRW and high-temperature damage in field experiments, and found that CIRW effectively

K. Nishida et al. / Agricultural Water Management 198 (2018) 10–18

reduces heat damage to rice. On the other hand, it has been pointed out that applications of these non-conventional water management methods such as CIRW inevitably affect water demand (Tomosho and Yamashita, 2009). In particular, CIRW requires more irrigation water than do conventional water management methods (Tomosho and Yamashita, 2009). Thus, applying CIRW without considering the water supply may cause water-use problems. For the practical application of CIRW, it is important to establish a theoretical basis for optimum water management conditions (e.g. irrigation rate and amount, timing, water depth, duration), which minimize water consumption but maximize the effect on the thermal environment. Analyses based on the physical mechanism of the formation of the thermal environment in a paddy field will provide much information on this subject. Previously, substantial studies on the heat environment of paddy fields have been conducted, and models to describe the heat environment have been developed (e.g. Inoue, 1985; Kim et al., 2001; Kondo and Watanabe, 1992; Saptomo et al., 2004). These models have been widely used and have been shown to describe the heat transfer of paddy fields with high accuracy. However, these models do not always take into account the effects of irrigation, and assume that there is horizontally uniform water and vegetation temperature distribution inside the paddy field. Thus, they cannot represent the horizontally varying water temperatures that have been reported in previous studies (e.g. Miyasaka et al., 2011; Nishida et al., 2013; Tsuboi and Honda, 1953). Because the cool irrigation water is heated via heat exchange with the atmosphere, rice canopy, and soil as it flows through the paddy field, it is inevitable that a temperature gradient will form along the longitudinal line of the field. Moreover, the temperature distribution differs according to the irrigation rate and water depth (Nishida et al., 2013, 2015). Thus, to evaluate the effect of CIRW on the heat environment of a paddy field, not only the heat exchange with the atmosphere, but also the effect of the heat convection driven by the inflow of irrigation water must be taken into account. However, these parameters have been considered in only a few studies. Thus, quantitative analyses of the effects of water management on the heat environment in paddy fields are insufficient. The purposes of this study were as follows: 1) to derive theoretical equations to represent the water and vegetation temperatures in a paddy field; and 2) to analyze the effects of irrigation conditions (irrigation rate and water depth) on water and vegetation temperatures by using the obtained equations to evaluate the effects of the water management methods such as CIRW on the temperature environment of the paddy field. First, explicit formulae that were able to express not only the effects of atmospheric conditions, but also those of irrigation conditions on the temperatures of paddy water and vegetation (irrigated rice), were derived by analytically solving the heat balance equations of the water and the vegetation, including the effects of heat convection driven by irrigation. The obtained equations were validated by comparisons with observed water temperature distribution in a farmer-managed paddy field under CIRW conditions. Then, various calculations were conducted using these equations to analyze the effects of irrigation conditions on the water and vegetation temperatures.

2. Governing equations of heat balance of paddy water and vegetation A double-source model (Kondo and Watanabe, 1992), which considers the heat balance at ground and vegetation separately, is among the widely accepted meteorological models to express heat exchange between atmosphere and soil. This model has been applied to paddy fields in many previous studies and estimates heat transfer with high accuracy (Kondo and Watanabe, 1992;

11

Maruyama and Kuwagata, 2008, 2010). However, because the original double-source model of Kondo and Watanabe (1992) does not take into account the horizontal heat convection in a paddy field, it cannot estimate the effect of CIRW on the thermal environment. We started by introducing a heat convection term into the doublesource model. 2.1. Heat balance equations To simplify the analysis, we assumed that water flow in a paddy field is horizontal and one dimensional from the irrigation inlet through to the outlet of the paddy field; that water depth and irrigation rate are constant with time; and that there is no infiltration. In general, CIRW uses much more irrigation water than conventional water management (Tomosho and Yamashita, 2009), thus, irrigation rate under CIRW condition is much higher than infiltration rate. Moreover, CIRW irrigates water and simultaneously drains the paddy water to allow the irrigation water to replace the paddy water, thus water depth under CIRW condition gradually becomes constant. Thus, the assumptions of no infiltration and constant water depth are reasonable under CIRW condition. Under these assumptions, the heat balance equations of paddy water (Eq. (1)) and the rice canopy (Eq. (2)) are as follows: 4 fv R↓ + (1 − fv )Tc4 = Tw + Hw + LEw + G + Cw h

∂Tw ∂Tw + Cw Q ∂t ∂x (1)

4 ) = 2(1 − fv )Tc4 + Hc + LEc . (1 − fv )(R↓ + Tw

(2)

Where Tc is the mean vegetation temperature of rice (K), Tw is the paddy water temperature (K), R↓ is incoming radiation including solar radiation ((1-albedo) × solar radiation) and longwave radiation from the sky (W m−2 ), fv is the transmissivity of the vegetation for radiation, ␴ is the Stefan-Boltzmann constant (5.67 × 10−8 W m−2 K−4 ), Hw and Hc are the sensible heat flux (W m−2 ) of paddy water to the atmosphere and vegetation to the atmosphere, respectively, LEw , and LEc are the latent heat flux (W m−2 ) of paddy water to the atmosphere and vegetation to the atmosphere, respectively, G is downward heat flux from paddy water to the soil (W m−2 ), Cw is the volumetric heat capacity of water (4.184 × 106 J m−3 K−1 ), h is water depth in the paddy field (m), t is time (s), Q is the water flow rate per unit width in the paddy field (m2 s−1 ), and x is the horizontal distance from the inlet of the paddy field (m). The left side terms in both of the equations are the incoming radiation to the paddy water and the vegetation, respectively. The last two terms in Eq. (1) are those newly added to the original double-source model to express the effect of CIRW. The theoretical equations of Tw and Tc under CIRW can be obtained by solving Eq. (1) and Eq. (2), respectively. 2.2. Heat fluxes The sensible heat fluxes (Hw and Hc ), the latent heat fluxes (LEw and LEc ), and the downward heat flux G are expressed by following equations: Hw = KHW (Tw − Ta )

(3)

Hc = KHc (Tc − Ta )

(4)

LEw = kLEW (qsat (Tw ) − qa )

(5)

LEc = kLEc (qsat (Tc ) − qa )

(6)

∂Ts | ∂z z=0

(7)

G = −

12

K. Nishida et al. / Agricultural Water Management 198 (2018) 10–18

Where Ta is the air temperature (K), qa is the specific humidity of air (kg kg−1 ), qsat (Tw ) and qsat (Tc ) are the specific humidity at the saturation-specific humidity (kg kg−1 ) at a temperature of Tw and Tc , respectively, Ts is the soil surface temperature (K),  is the heat conductivity of soil (W m−1 K−1 ), and z is soil depth (m). KHw and KHc denote sensible heat exchange coefficients from the paddy water to the atmosphere and from the vegetation to the atmosphere, respectively. kLEW and kLEc are the latent heat exchange coefficients from paddy water to the atmosphere and from vegetation to the atmosphere, respectively. Because the heat exchange coefficients are proportional to bulk transfer coefficients and wind speed, they are given as follows (Kondo and Watanabe 1992): KHW = ca a CHw u

(8)

KHc = ca a CHc u

(9)

kLEW = la CEw u

(10)

kLEc = la CEc u

(11)

Where ca is the specific heat of the air (J kg−1 K−1 ), ␳a is the density of the air (kg m−3 ), l is the latent heat of evaporation (J kg−1 ), CHw and CHc are the bulk transfer coefficients for sensible heat flux between the paddy water and the atmosphere and between the vegetation and the atmosphere, respectively, and CEw and CEc are the bulk transfer coefficients for latent heat flux between the paddy water and the atmosphere and between the vegetation and the atmosphere, respectively. The values of the bulk transfer coefficients generally depend on both the roughness lengths and the thermal stability of the airflow. However, including thermal stability makes it difficult to solve equations analytically because of the non-linearity of the term expressing the thermal stability effect. Thus, we simplified that the stability of the atmosphere is neutral. In this case, the coefficients are independent of Tw and Tc . The four bulk transfer coefficients of the atmosphere in neutral conditions in the doublesource model can be calculated by the following equations from Watanabe (1994): CHw

k2 = ln[(za − d)/z0w ] ln[(za − d)/zTw ]

CHc = CH − CHw CEw

k2 = ln[(za − d)/z0w ] ln[(za − d)/zqw ]

CEc = CE − CEw

(12)

In this study, the following values obtained by Kimura and Kondo (1998) were used: z0s = 0.001 m, zTs = 0.001 m, zqs = 0.001 m, cd = 0.2, and ch = 0.06. 3. Analytical solutions of paddy water and vegetation temperature 3.1. Linearization of flux equations The heat balance equations of Eqs. (1) and (2) were non-linear equations of Tw and Tc because of the existence of terms for latent heat and long-wave radiation. To obtain the analytical solution, the nonlinear terms in Eqs. (1) and (2) were linearized as described below. The linear forms of latent heat fluxes with respect to temperature are as follows: LEw = kLEW (qs (Tw ) − qa ) ∼ = kLEW

∂qs | (Tw − Ta ) ∂T Ta (18)

+kLEW (1 − RHa )qs (Ta ) = KLEW (Tw − Ta ) + kLEW (1 − RHa )qs (Ta )

LEc = kLEc (qs (Tc ) − qa ) ∼ = kLEc

∂qs | (Tc − Ta ) + kLEc (1 − RHa )qs (Ta ) ∂T Ta

= KLEc (Tc − Ta ) + kLEc (1 − RHa )qs (Ta ) (19) where KLEW = kLEW KLEc = kLEc

∂qs | ∂T Ta

∂qs | ∂T Ta

(20)

(21)

and RHa is the relative humidity of air. The long-wave radiation terms are linearized by Taylor’s expansion as follows:

(13)

4 ∼ Tw = Ta4 + 4Ta3 (Tw − Ta ) = 4Ta3 Tw − 3Ta4

(22)

(14)

Tc4 ∼ = Ta4 + 4Ta3 (Tc − Ta ) = 4Ta3 Tc − 3Ta4

(23)

(15)

The soil conduction flux is expressed by the method of Bhumralkar (1975) (force restore method) as follows:

where k is the Karman constant (0.4), za is the reference height (m), and d is the zero-plane displacement height (m). z0w , zTw , and zqw are the roughness lengths (m) that express the effects of the water surface on profiles of wind speed, temperature, and specific humidity above the vegetation, respectively. CH and CE are the bulk transfer coefficients for sensible and latent heat exchange between the entire surface and atmosphere, respectively, and are described as follows: CE =

k2 ln[(za − d)/z0 ] ln[(za − d)/zq ]

(16)

CH =

k2 ln[(za − d)/z0 ] ln[(za − d)/zT ]

(17)

where z0 , zT , and zq are the roughness lengths (m) of the entire surface for profiles of wind speed, temperature, and specific humidity above the canopy, respectively. The seven parameters d, z0 , zT , zq , z0w , zTw , and zqw can be calculated from the empirical equations of Watanabe (1994) with the values of height of canopy hc , leaf area index LAI, leaf transfer coefficients cd, ch, roughness lengths of the ground surface z0s , zTs , zqs , and bulk stomatal conductance, gs .

G = Cs ds

∂Tw + Ks (Tw − Tw ) ∂t

(24)

 ) 2Cs ω

(25)

where



ds =

(

 Ks =

Cs ω 2

(26)

where Cs is the heat capacity of soil (J m−3 K−1 ), w is the frequency of oscillation (=2␲/86400 s−1 ), and Tw is the daily mean water temperature (K). ds is the imaginary soil depth that has the same temperature as that of the paddy water and Ks is the heat exchange coefficient between the paddy water and deep soil, whose temperature is assumed to be Tw . Thus, the first and second terms of Eq. (24) can be considered as the heat storage of surface soil and the heat conduction from the paddy water to the deep soil, respectively.

K. Nishida et al. / Agricultural Water Management 198 (2018) 10–18

3.2. Relationship between vegetation temperature and paddy water temperature Assigning the linearized flux equations of Eqs. (19), (22) and (23) to Eq. (2), the linearized heat balance equation of Eq. (2) is as follows: (1 − fv )(R↓ + (4Ta3 Tw − 3Ta4 )) = 2(1 − fv )(4Ta3 Tc − 3Ta4 )

(27)

+KHc (Tc − Ta )

boundary condition of x = 0 is irrigation water with a constant temperature (Ti ). The length between the inlet and the outlet of the paddy field is given as L (m). Since the water depth h and water flow rate in the paddy field Q are assumed to be constant under these conditions, the relationship between irrigation rate Qin (m/s) and Q is expressed as Q = Qin × L. By assigning the linearized flux equations of Eqs. (18), (22)–(24), and (28) to Eq. (1), the heat balance equation can be written as a partial differential equation, as follows: (Cw h + Cs ds )

+KLEc (Tc − Ta ) + kLEc (1 − RHa )qs (Ta ) By solving Eq. (27) for Tc , Tc can be written in the linear equation for Tw as follows: Tc = aTc Tw + bTc

(28)

where, aTc = bTc =

(29)

8(1 − fv )Ta3 + KHc + KLEc (1 − fv )(R↓ − 5Ta4 ) − kLEc (1 − RHa )qs (Ta ) + KHc + KLEc

+ Ta .

(30)

Eq. (28) indicates that the changes in Tw , Tw , provide the changes in Tc of aTc × Tw . Because KHC and KLEC do not depend on Tw , the coefficient of aTc is determined only by atmospheric conditions. Using Eq. (28), Tc can be obtained from Tw , making it possible to evaluate the effects of various water management methods such as CIRW and changing water depth on leaf temperature. 3.3. Estimation of water temperature under no-irrigation conditions

(0 ≤ x ≤ L) (0 < t < ∞)

(35)

T w (0,t) = T i (0 < t < ∞)

(36)

T w (x, 0) = T w0 (0 ≤ x ≤ L)

(37)

dTw (t) ∞ = −aTw (Tw (t) − Tw ) dt

(31)

where aTw = (4Ta3 + KHW + KLEW + Ks − (1 − fv )4Ta3 aTc )

(32)

∞ = {f R + f 3T 4 Tw v ↓ v a

−kLEW (1 − RHa )qs (Ta ) + (KHW + KLEW )Ta + Ks Tw

(33)

+(1 − fv )4Ta3 bTc }/aTw Given that aTw and Tw ∞ are independent of Tw , assuming atmospheric conditions are constant with time, Eq. (31) can be solved by separation of variables. Under initial conditions of Tw = Tw (t0 ), ∞ ∞ Tw (t) = Tw + (Tw (t0 ) − Tw ) exp(−

aTw (t − t0 )). Cw h + Cs ds

vT = Cw Q/(Cw h + Cs ds )

(34)

Eq. (34) indicates that the water temperature gradually converges to Tw ∞ , which can be regarded as the equilibrium water temperature determined by atmospheric and soil conditions.

s = x − vT t

(39)

t=t

By using this transformation, Eq. (35) becomes the same form as Eq. (31) as follows: dTw (s, t) ∞ ) = −aTw (Tw (s, t) − Tw dt

The situation of irrigation (heat convection) was considered under the following conditions: initial paddy water temperature Tw0 is assumed to be the same inside the paddy field; and the

(40)

Equation (40) can be solved by the same method described in Section 3.3. Thus, the water temperature Tw (s,t) under initial conditions of Tw (s,t0 ) at t = t0 is written as follows: ∞ ∞ Tw (s, t) = Tw + (Tw (s, t0 ) − Tw ) exp(−

aTw (t − t0 )). Cw h + Cs ds

(41)

Transforming Eq. (41) to that with x and t (Tw ) coordinates can be written as follows: aTw ∞ ∞ Tw (x, t) = Tw + (Tw (x − vT t, t0 ) − Tw ) exp(− (t − t0 )).(42) Cw h + Cs ds The water temperature of Tw (x-vT t, t0 ) in Eq. (42) varies with the values of the x and t coordinates; two cases of Tw (x-vT t, t0 ) can be considered. Fig. 1 shows characteristic lines of the solutions of Tw (x,t) on an x,t plane. By backtracking along the characteristic lines on Fig. 1, Tw (x-vT t, t0 ) can be found. Note, since we assume no infiltration and constant water depth, vT in Fig. 1 is constant in the paddy field. But, when the infiltration rate and changes in water depth can not be neglected, vT decreases with increase in distance from the inlet of paddy field. When x is longer than vT t as shown in A in Fig. 1, there is no effect of heat convection from irrigation water. In this case, Tw (x-vt, t0 ) is the initial water temperature of the paddy field (Tw0 ): T w (x − vT t,t 0 ) = T w0 .

(43)

In this case, Tw (x,t) is ∞ ∞ Tw (x, t) = Tw + (Tw0 − Tw ) exp(−

3.4. Solution of water temperature during irrigation

(38)

Eq. (38) indicates that the velocity of heat conduction vT is slower than that of water v (=Q/h) because of the effect of the heat storage of soil. Using vT , Eq. (35) can be changed to an ordinary differential equation by a coordinate transformation as follows:

(Cw h + Cs ds )

The equation to calculate the paddy water temperature when there is no inflow of irrigation water can be derived by assigning Q = 0 to Eq. (1). Without water convection, water temperature, Tw , does not depend on space, only on time. Thus, the linearized heat balance equation can be written as an ordinary differential equation (Eq. (31)) by introducing the linearized flux Eqs. (18) and (22)–(24) and the leaf temperature equation of Eq. (28) into Eq. (1) as follows: (Cw h + Cs ds )

∂Tw (x, t) ∂Tw (x, t) ∞ ) − Cw Q = −aTw (Tw (x, t) − Tw ∂t ∂x

Here, we define the heat convection velocity vT (m s−1 ) as follows:

4(1 − fv )Ta3

8(1 − fv )Ta3

13

aTw t). Cw h + Cs ds

(44)

Similar to Eq. (34), Eq. (44) indicates that the water temperature gradually converges to the equilibrium temperature. In this case, water temperature is affected by water depth but not irrigation rate.

14

K. Nishida et al. / Agricultural Water Management 198 (2018) 10–18

Fig. 2. Schematic representation of observed paddy field showing measurement locations.

Fig. 1. Characteristic line of water temperature in a paddy field in the xt-plane. Tw (x,t) is water temperature, x is distance from paddy field inlet, t is time after the start of irrigation, vT is heat convection velocity, and L is the paddy length. Tw0 is water temperature at the start of irrigation, Ti is irrigation water temperature.

On the other hand, when x is less than vT t as shown in B in Fig. 1, Tw (x0 -vt0, t0 ) describes the irrigation water temperature (Ti ): T w (x − vT t,t 0 ) = T i .

t0 = t −

vT

.

(47)

4.2. Calculation method

Substituting Eqs. (45) and (47) into Eq. (42) yields the following equation: ∞ ∞ Tw (x, t) = Tw + (Ti − Tw ) exp(−

aTw x) Cw Q

(51)

(46)

In this case, the relationship between x and t0 is expressed as follows:

x

Qin = h + I + Qout (h)

Where, Qin is the irrigation rate (mm/h), h is the increase in water depth (mm/h), I is the infiltration rate (mm/h), Qout (h) is the surface runoff from the outlet of paddy field (mm/h). I was determined from changes in water depth during nighttime with no irrigation and no surface runoff. Qout (h) was calculated from water depth using the relationship between water depth and surface runoff at the outlet (Nishida et al., 2016).

(45)

x = vT (t − t0 )

profiles and water depth were monitored. The horizontal distributions of water temperature and irrigation water temperature were obtained by thermometers (KN laboratories. Inc., Osaka, Japan) set at nine positions along the longitudinal line and at the irrigation canal (Fig. 2). The changes in water depth were measured by a water pressure gauge. Measurements were taken at 10-min intervals. The irrigation rate was calculated from the water balance of the paddy field (Eq. (51)) using measured water depth change, an infiltration rate (0.45 mm h−1 ) and surface runoff from the outlet of paddy field.

(48)

Equation (48) indicates that the water temperature gradually reaches equilibrium temperature with as the distance from the irrigation water inlet increases, and is not affected by the initial water temperature (Tw0 ). Vegetation temperature can be also evaluated by substituting Eqs. (44) and (48) into Eq. (28). We defined relaxation time ␶ (s) and relaxation length ␰ (m) as follows: =(

−1 aTw ) Cw h + Cs ds

(49)

=(

aTw −1 ) Cw Q

(50)

 and ␰ are the useful indicators of temporal and spatial changes in water temperature:  is the time during which the changes in water temperature become 1/e, and ␰ is the distance at which the changes in water temperature become 1/e. 4. Validation of analytical model 4.1. Monitoring of water temperature in a rice paddy field The water temperature was monitored during the rice ripening period in August 2013, at a farmer-managed paddy field in Ishikawa Prefecture, Japan (Nishida et al., 2016). The field was rectangular (26 m × 78.5 m) with an irrigation water inlet and outlet at both ends of the longer side of the field (Fig. 2). CIRW was used by the farmer during the nighttime, and changes in water temperature

Because the solutions obtained in this study hold true under constant atmospheric conditions, the time series of the solution should be calculated for each time interval in which the atmospheric conditions can be regarded as constant. Thus, in this study, we numerically calculated Tw by using a discretized equation of Eq. (42). ∞ ∞ Tw j = Tw + (Tw 0j−1 − Tw ) exp(− j−1 j−1

aTw

t) Cw h + Cs ds

(52)

Where t is the time step, subscript j indicates node number (j = 0 is the inlet of irrigation), and superscript 0 indicates the value of the previous time step. The interval between the spatial nodes x is given as the distance that the heat convection front advances in

t;

x = vt t

(53)

By using this x, the water temperature of nodes i can be calculated from the previous time step value of node i-1. The distribution of water temperature in the field from August 12–13, 2013, was simulated. Fig. 3 shows the measured water depth and irrigation rate. Because a rapid change in water depth occurred a few hours after the start of irrigation, the simulation was performed for 10 h from 8:00 PMto 6:00 AM during which time the water depth and the irrigation rate were assumed to be constant (Fig. 3). The conditions for the simulation were as follows: the time step

t was given as 10 min; the water depth was set to 0.028 m, which was the measured water depth at the end of irrigation; and the irrigation rate was set to 7.7 mm h−1 (=1.68 × 10−4 m2 s−1 ), which was the average irrigation rate during the irrigation period (Fig. 3). Since the irrigation rate (7.7 mm/h) was 17 times higher than the infiltration rate (0.45 mm/h), infiltration rate was negligible. Thus,

K. Nishida et al. / Agricultural Water Management 198 (2018) 10–18

15

Fig. 3. Changes in water depth and irrigation rate in CIRW paddy field.

we assumed that vt was constant inside paddy field. Consequently,

x was 1.56 m and the total node number was 52. Irrigation water temperature was the measured value. Air temperature, relative humidity, wind speed, and solar radiation were measured values obtained at the experimental paddy field located at Isikawa Prefectural University, which is about 10 km from the paddy field. Downward long-wave radiation from the sky was estimated from the air temperature and humidity. The initial water temperature and daily average water temperature in Eq. (33) of each node were estimated from the fitting equation obtained by the measured water temperature distributions. Water temperatures at 20:00 August 12 were approximated by the quadratic function of x. Average water temperatures during the 24 h before 20:00 August 12 were approximated by the linear function of x. 4.3. Results The calculated and measured water temperatures as a function of time and distance from the inlet are shown in Fig. 4(a) and (b), respectively. The simulation successfully represented the increase in water temperature along the flow path in the field, consistent with results published in previous reports (e.g. Miyasaka et al., 2011; Nishida et al., 2013, 2015; Tsuboi and Honda 1953). The simulation also revealed a gradual decrease in water temperature over time. The calculated change in the spatial distribution of the water temperature over time showed good agreement with the measured changes. The root mean square error (RMSE) for the estimation of water temperatures was 0.39 ◦ C. From these results, we concluded the proposed model is capable of expressing qualitative and quantitative features of the paddy water temperature environment under CIRW conditions.

Fig. 4. Calculated and observed temperature values. (a) Changes in water temperature and (b) distribution of water temperature in CIRW paddy field. Plots: observed values. Lines: calculated values. Table 1 Parameters used for the simulation. (a) Meteorological conditions Reference level za : 2 m above soil surface Air temperature Ta : 25 ◦ C Relative humidity RHa : 80% Wind speed u: 0.5 ms−1 Downward long-wave radiation: 400 Wm−2 Solar radiation: 0 (nighttime) and 600 (daytime) Wm−2 (b) Soil and vegetation properties Surface albedo: 0.2 Heat capacity of soil Cs : 3.43 × 106 J m−3 K−1 Heat conductivity of soil ␭: 1 W m−1 K−1 Vegetation height hc : 1 m Leaf area index LAI: 4 Transmissivity of vegetation for radiation fv : 0.165 (=exp(−0.45 × LAI)) Stomatal conductance gs: 0 (nighttime) and 0.02 (daytime) (c) Paddy field length: 100 m.

temperatures in a CIRW paddy field. The parameters used for the calculations are summarized in Table 1. 5.1. Heat convection velocity

5. Evaluations of the effect of irrigation conditions on temperature environment of paddy field The equations obtained in our study are characterized by the parameters of vt, , ␰, Tw ∞ , and aTc . These parameters are affected by irrigation and meteorological conditions. Thus, analyzing the relationships between these conditions and the parameters is useful to understand and estimate the effect of CIRW on the temperature environment in a paddy field. In this section, examples of the relationships between certain parameters and irrigation and meteorological conditions are provided. These analyses allowed us to explore the effects of various conditions on water and vegetation

The heat convection velocity of vt (Eq. (49)), which is a function of irrigation rate and water depth, determines the distance over which the paddy water is affected by the cool irrigation water. Thus, the value of vt is useful for evaluating the extent of the cooling effect of CIRW. Fig. 5 illustrates the relationships between vt and (a) irrigation rate and (b) water depth. At the same water depth (Fig. 5 (a)), vt linearly increases with increasing irrigation rate; e.g. at 3-cm water depth with irrigation rates of 3, 5 and 10 mm/h, the values of vt are 4.5, 7.5, and 15.0 m/h, respectively. Under the same irrigation rate, vt increases with shallower water depth; e.g. at an irrigation rate of 5 mm/h with water depths of 3, 5, 10 cm, values of

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Fig. 6. Relationship between relaxation time  and water depth under different wind speeds, u.

Fig. 5. Relationship between heat convection velocity vt and (a) irrigation rate and (b) water depth.

vt are 7.5, 5.8, and 3.7 m/h, respectively. Increasing irrigation rate and shallower water depth both increase vt , so that the cooling effect of CIRW increases. Thus, CIRW with a high irrigation rate and shallower water depth is preferable to increase the cooling effect of CIRW. However, supplying large quantities of irrigation water is typically difficult because of limited water supply. Thus, application of CIRW in shallow water depths is likely to be more practical than supplying large quantities of irrigation water. 5.2. Relaxation time and relaxation length The relaxation time,  (Eq. (49)), and the relaxation length, ␨ (Eq. (50)), determine the rate of water temperature change over time and space, respectively. For example, when  and ␰ are doubled, the time and space required to change a unit of water temperature are halved. From Eq. (49),  linearly increases with increasing water depth. Fig. 6 illustrates the relationships between  and water depth during the nighttime under six wind speed conditions. For instance, when the wind speed is 0.5 m/s, values of  with water depths of 0, 1, 3, 5, and 10 cm are 2.5, 3.2, 5.3, 6.9, and 10.8 h, respectively. This result means that the time required to change a unit of water temperature with water depths of 1, 3, 5, and 10 cm are 1.3-, 1.8-, 2.4-, and 3.7-times that with a water depth of 0 cm. On the other hand, from Eq. (50), ␰ linearly increases with increasing irrigation rate. Fig. 7 illustrates the relationships between ␰ and irrigation rate during the nighttime under six wind speed conditions. For instance, when the wind speed is 0.5 m/s, values of ␰ with irrigation rates of 3, 5 and 10 mm/h are 20.5, 34.2, and 68.4 m, respectively. From the definition of Eq. (50), ␰ is determined only by the irrigation rate; thus, differences in the water depth and heat convection velocity do not affect ␰ when the irrigation rate is the same. A decrease in water depth increases the distance that heat convection advances within a unit of time because of the increase in

Fig. 7. Relationship between relaxation length ␰ and irrigation rate under different wind speeds, u.

vt . On the other hand, a decrease in water depth increases the rate of change in water temperature because of the decrease in the heat capacity of paddy water. Because these two mechanisms balance out, spatial changes in water temperature are determined only by the irrigation rate. This feature is quite different from the feature of vt . Both  and ␰ are affected by wind speed. Because the parameter of aTw in Eq. (49) and (50) is an increase function of wind speed,  and ␰ decrease with increasing wind speed. For example, when the water depth is 3 cm, values of  with wind speeds of 0, 0.5, 1, and 5 m/s are 5.3, 4.6, 4.2, and 2.9 h, respectively (Fig. 6). When the irrigation rate is 5 mm/h, values of ␰ with wind speeds of 0, 0.5, 1, and 5 m/s are 39.6, 34.2, 32.2, and 21.5 m, respectively (Fig. 7). An increase in wind speed increases the latent and sensible heat exchange coefficients of Eq. (8) and (9) so that the heat exchange by latent and sensible heat from paddy water accelerates. Thus, an increase in wind speed shortens the time and the distance when and where the water temperature will change. This result indicated that low wind speeds are preferable to high wind speeds for the cooling effect of CIRW. 5.3. Equilibrium water and leaf temperatures The equilibrium water temperature Tw ∞ defined by Eq. (33) determines whether water temperature increases or decreases with time and with distance from the inlet. Comparing the value of Tw ∞ and the temperatures of paddy and irrigation water provides information to determine the optimal water management to control the water temperature in a paddy field. For example,

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Fig. 8. Relationship between equilibrium water and leaf temperature and wind speed u. Tw: equilibrium water temperature, Tc: equilibrium leaf temperature, Ta: air temperature. ∞

when there is no irrigation, when Tw is similar to the intended or optimal water temperature compared to the initial water temperature, shallower water depths accelerate the change in water temperature. However, when Tw ∞ differs from the intended or optimal water temperature, deeper water depths inhibit the change in water temperature. In the case of CIRW, if the irrigation water temperature is higher than Tw ∞ , then the cooling effects of CIRW on paddy water become less pronounced. The equilibrium water and leaf temperatures are determined by meteorological conditions and soil temperatures. Fig. 8 illustrates the relationships between equilibrium water and leaf temperatures and wind speed during the daytime and nighttime, and also the air temperature. The equilibrium leaf temperatures are calculated by Eq. (28) by assigning Tw ∞ to Tw . The equilibrium water and leaf temperatures under low wind speed are mainly affected by solar and downward long-wave radiation, but under high wind speed, are mainly affected by air temperature and relative humidity. Accordingly, the equilibrium temperatures under low wind speed are lower (higher) during nighttime (daytime) than that under high wind speed. As shown in Fig. 8, Tw ∞ varies substantially depending on meteorological conditions. Thus, the optimal water management technique should change according to the meteorological conditions. 5.4. Coefficient of vegetation temperature Vegetation temperature is one of the most important factors in the heat damage of rice. The relationship between vegetation temperature and water temperature is characterized by the parameter aTc (Eq. (29)). The parameter aTc varies substantially with wind speed. Given that the transfer coefficients of sensible and latent heat fluxes from the leaf are linear with respect to wind speed, aTc of Eq. (29) can be rewritten as follows: aTc =

KLc /2 



KLc + (KHc + KLEc )u

(54)

where KLc = 8(1 − fv )Ta3 . Equation (54) indicates that aTc becomes 0.5 when the wind speed is zero, and decreases as the wind speed increases. Examples of the relationships between aTc and wind speed during the daytime and nighttime are shown in Fig. 9. As shown in Fig. 9, aTc decreases with increasing wind speed, and is higher at nighttime than during the daytime: For example, values of aTc under wind speeds of 0.5, 1, and 3 m s−1 are 0.24, 0.16 and 0.07, respectively, at nighttime, and 0.10, 0.06, and 0.02, respectively, during the daytime. These results indicated that a decrease in leaf temperature of 1 (K) under wind

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Fig. 9. Relationship between wind speed and leaf temperature coefficient aTc .

speeds of 0.5, 1, and 3 m s−1 at nighttime require decreases in paddy water temperature of 1/0.24, 1/0.16, and 1/0.07 (K), respectively. Leaf temperature is typically determined by heat exchange (radiation, sensible heat, and latent heat) with the atmosphere and surface water. In particular, the long-wave radiation flux from paddy water must be affected by CIRW with cool water, because this flux is dependent only on paddy water temperature. Decreases in water temperature induced by CIRW with cool water reduce this flux, resulting in lower leaf temperature. On the other hand, increases in wind speed increase the diffusion of turbulence, increasing the effects of sensible and latent heat fluxes on leaf temperature. In such cases, leaf temperature is determined primarily by the air temperature; thus, when the air temperature is high, increases in wind speed decrease the cooling effect of CIRW. On the other hand, the effect of water temperature on rice temperature decreases during the daytime, because latent heat flux exists during the daytime. Thus, aTc and the cooling effect of CIRW on leaf temperature are smaller during the daytime than at nighttime under the same water temperature conditions, as shown in Fig. 9. These results indicated that the cooling effect of CIRW is effective under weak wind conditions and during the nighttime. 6. Conclusion In this study, equations to calculate the water and vegetation temperatures in a paddy field under horizontal water flow were obtained, and then used to evaluate the effect of CIRW on the temperature environment of the paddy field. The equations calculated the effect of irrigation conditions on changes in water and leaf temperature distribution under various meteorological, soil, and irrigation conditions. The equations were validated by comparisons with measured values obtained in a CIRW paddy field. The spatial and temporal changes in the calculated water temperatures showed good agreement with observed values. Therefore, the equations where shown to accurately represent the temperature environment in paddy fields under different water management conditions. The effects of irrigation rate, paddy water depth and wind speed on water and leaf temperatures were theoretically analyzed using the equations. The following results were obtained. 1) The area cooled by CIRW was positively related to irrigation rate and negatively related to water depth. Thus, CIRW is effective for high irrigation rates and shallow water depths. Shallow water depths, in particular, are preferable for application of CIRW. 2) Water temperature distribution within the heat convection front under CIRW was affected by the irrigation rate, but not by water depth. 3) Higher wind speed and latent heat flux from leaves decreased the effect of water temperature on leaf temperature. Thus, the cooling of leaves

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by CIRW is effective under weak wind conditions and during the nighttime. These results are the common features of the effects of irrigation on the temperature environment. Thus, the proposed equations and results are useful to evaluate the efficacy of water management techniques in controlling the heat environment of paddy fields. Acknowledgments This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology through the Research Program on Climate Change Adaptation (RECCA) entitled “Development of Decision Support System for Optimal Agricultural Production under Global Environment Changes.” and by JSPS KAKENHI Grant Number JP16K07939. We thank Mr. I. Yoshikawa, the owner of the rice paddy field where this study was conducted. References Arai, Y., Ito, H., 2001. Effect of flow irrigation on high temperature ripening in paddy field rice. Tohoku J. Crop Sci. 44, 89–90 (in Japanese). Arai, M., Matsumoto, Y., Arakawa, I., 2006. Milky white rice kernels prevention using transplantation time shift and water management. Tohoku Agric. Res. 59, 19–20 (in Japanese). Bhumralkar, C.M., 1975. Numerical experiments on computation of ground surface-temperature in an atmospheric general circulation model. J. Appl. Meteorol. 14, 1246–1258. Inoue, K., 1985. A simulation model for micrometeorological environment in rice field. J. Agr. Met. 40, 353–360 (in Japanese with English abstract). Kim, W., Arai, T., Kanae, S., Oki, T., Musiake, K., 2001. Application of the simple biosphere model (SiB2) to a paddy field for a period of growing season in GAME-Tropics. J. Meteorol. Soc. Jpn. 79, 387–400. Kimura, R., Kondo, J., 1998. Heat balance model over a vegetated area and its application to a paddy field. J. Meteorol. Soc. Jpn. 76, 937–953. Kondo, J., Watanabe, T., 1992. Studies on the bulk transfer-coefficients over a vegetated surface with a multilayer energy budget model. J. Atmos. Sci. 49, 2183–2199. Maruyama, A., Kuwagata, T., 2008. Diurnal and seasonal variation in bulk stomatal conductance of the rice canopy and its dependence on developmental stage. Agric. For. Meteorol. 148, 1161–1173. Maruyama, A., Kuwagata, T., 2010. Coupling land surface and crop growth models to estimate the effects of changes in the growing season on energy balance and water use of rice paddies. Agric. For. Meteorol. 150, 919–930. Ministry of Agriculture, 2008. Forestry and Fisheries of Japan, http://www.maff.go. jp/j/press/seisan/engei/pdf/080418-01 (Accessed November 2016).

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