Introduction Description of the model

Agricultural and Forest Meteorology 106 2001 131–146 Development and validation of model for estimating temperature within maize ear S. Khabba a,∗ , J.-F. Ledent b , A. Lahrouni a a Département de physique, Faculté des Sciences Semlalia, BP 2390, Marrakech, Morocco b ECOP Grandes Cultures, Université Catholique de Louvain, 2 pl. de la Croix du Sud, B-1348 Louvain-la-Neuve, Belgium Received 21 January 2000; received in revised form 18 July 2000; accepted 31 July 2000 Abstract We present a three-dimensional computer model that simulates ear temperatures under field conditions for both daytime and night-time. The meteorological data used are total and diffuse radiation, wind speed, air temperature and humidity or wet bulb temperature. The model is based on the energy variation of volume elements on ear surface. It takes into account, net radiation, sensible and latent heat exchange and heat diffusion within the ear. The model performs a radiation balance that separates direct, diffuse and scattering components. The husk stomatal resistance was parameterised as a function of water vapour deficit and solar radiation deduced from our experimental data. The model was tested in two stages: first, the calculated flux of downward and upward all wave radiation, at ear level, was compared with real measurements. Second, the calculated grain temperatures were compared with air temperature, and with data collected, for different polar positions around the cob, in two experiments conducted in 1997, in Morocco, and 1998, in Belgium. The agreement was satisfactory; the average difference between the model estimates and measurements of grain temperature were 0.5 ◦ C in Belgium and 0.6 ◦ C in Morocco, whereas using air temperature as the simplest estimate of the grain temperature gave average differences against the measured grain temperature of 1.1 and 1.8 ◦ C, respectively. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Maize; Ear; Temperature; Model

1. Introduction

Temperature has a major influence on plant develop- ment, growth and yield of maize Miedema, 1982. In maize, heat stress, around flowering, may be the cause of unsuccessful fertilisation with losses of 30–32 in grain yield Saadia et al., 1996. During grain filling, low temperatures affect grain growth Ledent, 1988 possibly due to effects on the transfer of assimilates through the cob to the grains. Grain temperature may ∗ Corresponding author. Tel.: +212-4-43-46-49; fax: +212-4-43-74-10. E-mail address: khabbaucam.ac.ma S. Khabba. also affect characteristics of seed quality Rossman, 1949. These effects are determined by temperature within maize grains which may differ significantly from the surrounding air temperature especially when the latter changes rapidly. The ear has thermal iner- tia Ledent, 1988; Khabba et al., 1999a, and the husk leaves protect the grain from variations of external temperature. Thus, a gradient of temperature may ex- ist between kernel and air surrounding the ear Ledent, 1988. Studies of the thermal behaviour of the ear have been done e.g. Woodams and Nowrey, 1968; Polley et al., 1980; Singh, 1982; Ledent et al., 1993; Khabba et al., 1999a. There have been attempts to model in- 0168-192301 – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 1 9 2 3 0 0 0 0 2 0 1 - X 132 S. Khabba et al. Agricultural and Forest Meteorology 106 2001 131–146 ternal temperature Gaffney et al., 1980; Di Pentima and Güemes, 1987; Khabba et al., 1995 but these were not done under field conditions, with ears attached to the plants within the plant canopy. We propose here a simple three-dimensional model, based on the energy balance of each elementary vol- ume on the ear surface and on heat conduction within the ear, to predict ear temperatures under field condi- tions from meteorological observations.

2. Description of the model

Internal ear temperature may be important when variations of air temperatures are rapid. Extreme dam- aging temperatures may last only short periods. We thus estimated ear temperature using a small step time 1 min. The ear is simulated as an inclined cylinder, consisting of three concentric layers with a variable Fig. 1. Schematic presentation of longitudinal section of maize ear and representation of the short wave radiation balance of the ear. R b and R d are, respectively, direct and diffuse downward solar radiation measured on a horizontal plane; R bd , R dd and R sr are direct, diffuse and scattered radiation reaching ear surface; φ s and φ e are solar azimuth and ear azimuth, respectively; β is solar elevation. A, B, and C identify locations on the ear. cross-section, attached to the peduncle Fig. 1. The variation of the section radius along the ear generat- ing AB or CB was described by the parabolic equa- tion developed by Khabba et al. 1999a. Heat can be transferred from the stem to the ear by sap flow. A rough estimate indicates that this flux may represent only 2–4 of incident radiation. We then assumed that heat exchanges through the lower end of the ear AC was negligible. For high sun elevation angles, an important amount of direct beam radiation reaches the ear surface. Ker- nel temperature depends on the polar position of the grain around the cob Khabba et al., 2000, especially between 13 and 16 h UT. Therefore, the polar dis- tribution of direct solar radiation on the ear surface and three-dimensional diffusion of heat within the ear are important to be accounted for. As in Khabba et al. 1999a, heat transfer within the ear was de- scribed by Fourier’s law. Estimates of temperature at S. Khabba et al. Agricultural and Forest Meteorology 106 2001 131–146 133 external husk surface will be made with respect to the field conditions. The temperature of each elementary volume on the external husk surface Fig. 1, generating ABC was calculated using the energy conservation law Saatdjian, 1993: Energy input = Energy absorbed from solar reaction + Energy exchange by long wave radiation + Energy lost by convection + Energy lost by evaporation + Energy exchange by conduction into ear The first four variables of the right-hand term are the external boundary conditions on the husk surface. The nature of each variable is discussed in turn below. 2.1. Short wave radiation, R sw Eq. 1 describes net short wave radiation R sw , for each element of the external surface area of the ear, in terms of its components Fig. 1: R sw = 1 − a e R bd + R dd + R sr 1 where a e is the ear albedo, R bd the downward di- rect beam solar radiation W m − 2 normal to the ear surface, R dd the downward diffuse solar radiation Fig. 2. Schematic representation of the subdivision of the canopy: 1 horizontal layer; 2 vertical slice; 3 cell of vegetation. The canopy is divided into N z vertical layers and N x vertical slices between two rows parallel to the direction of the row. The canopy structure is assumed symmetric on both sides of row and on both sides of inter-rows line. W m − 2 and R sr the scattering radiation W m − 2 . The components of this equation are not measured directly, but estimates are made using measured weather data. 2.1.1. Description of the canopy In the case of the row crops, the canopy structure of the vegetation is divided into N z horizontal layers and N x vertical slices parallel to the direction of the row Fig. 2. The intersection of the slice and the layer gives a cell, k, of vegetation. The thickness of the layers was chosen to be of the same order of magnitude as ear length. Each ear is situated in one cell and this cell is referred by, e. Each cell, k, may be characterised by its leaf area densities a k and its leaf inclination distribution g k α random leaf azimuth distribution was assumed. 2.1.2. Radiation interception by vegetation For a given direction Ω i.e. height β and azimuth φ, φ = 0 ◦ for the row direction, the probability of non-interception in a cell k is given by the classical negative exponential law: P k = exp−K[g k α, Ω]a k δz 2 δz is the length of the trajectory of the beam inside the cell, it was calculated using classical trigonometri- cally lows Fukai and Loomis, 1976; Sinoquet, 1989. 134 S. Khabba et al. Agricultural and Forest Meteorology 106 2001 131–146 K[g k α, Ω] is the projection of an unit area of leaf, with an inclination distribution g k α, on the plane normal to Ω. K[g k α, Ω] has been described by sev- eral authors e.g. Ledent, 1977; De Castro and Fetcher, 1998. Nilson 1971 and Lemeur and Blad 1974 compared different functions for calculating the prob- ability of interception for distributions of foliage: ran- dom, regular or clumped. The negative exponential function used here assumes a random distribution of foliage within the cell. The probability that a beam light reaches the cell k, P ′ k , is expressed as P ′ k = P 1 , P 2 , . . . , P n 3 where n is the number of cells in the path of the beam to reach the cell of interest, k. 2.1.3. Direct radiation, R bd Direct radiation interception is computed from the above considerations applied to the sun direction Ω s . Hence, direct radiation reaching the ear cell e my be expressed as R bΩ s = R b P ′ e Ω s 4 where R b is the direct solar radiation flux density mea- sured on a horizontal surface, and P ′ e Ω s the mean probability of a beam of direction Ω s reaching the ear e.g. Allen, 1974; Fukai and Loomis, 1976; Sinoquet, 1989. At any time, a local and instantaneous value of R bd can be expressed as follows: R bd =    cos θ cot gβ + α cos1φR bΩ s if 0 ≤ θ ≤ π 2 , if π 2 ≤ θ ≤ π 5 β and Ω s can be easily calculated using classical as- tronomical formulae from the latitude of the site and the day of the year e.g. Garnier and Ohmura, 1968; De Castro and Fetcher, 1998. θ is the polar angle be- tween solar beam direction and the normal to the sur- face; 1φ = φ s − φ e is the difference between sun and ear azimuths, and α the ear inclination Fig. 1. Both φ s and φ e are calculated relative to the row direction. 2.1.4. Incident diffuse radiation, R dd Diffuse radiation comes from all directions with variable intensity depending on elevation of the radi- ation and other factors. It was treated as coming from a set of directional radiation sources, i.e. integrating contributions from the whole sky. Therefore, the sky was divided into solid angle sectors dΩ according to class of heights and azimuth angles. The amount of incident diffuse radiation R b Ω coming from each angle sector dΩ was derived from the Standard Over- Cast sky SOC distribution Moon and Spencer, 1942 or Uniform OverCast sky UOC distribution Walsh, 1961. The mean diffuse radiation reaching the ear may be written as R dd = nΩ X Ω=1 R d ΩP ′ e Ω 6 Summing over solid angle was performed with class intervals of 10 ◦ for β and 20 ◦ for φ. 2.1.5. Radiation scattering, R sr Maize ear is generally situated in the canopy at about half height. It can receive scattered radiation from all surrounding cells or soil strips. A precise com- putation of the flux density of scattering radiation, R sr , requires the use of a method treating all radiation ex- changes within the canopy. The radiosity method has been frequently used for that purpose Neveu, 1984. Its use is possible because the soil–vegetation–sky forms a closed system. The calculation of flux den- sity R sr was split into two steps. First, the flux R e intercepted by the cell of the ear was calculated using the method fully described by Sinoquet 1989 and Sinoquet and Bonhomme 1992. Second, R sr is de- duced from R e by R sr = R e − R be + R de 7 2.2. Long wave radiation, R lw For the sake of simplicity, we assumed that the emis- sivities of the soil, foliage elements and ear were equal to 1, and that their emitted radiation was isotropic. As- suming an emissivity of the soil surface equal to 0.9 would to lead an error less than 5 on the long wave radiation balance. The field of view of the ear is made S. Khabba et al. Agricultural and Forest Meteorology 106 2001 131–146 135 up of three different zones: the soil surface, the sur- rounding cells of vegetation and the atmosphere. The soil surface can reasonably be considered as having a uniform surface temperature T sol , but the flux density of long wave radiation coming from the atmosphere is strongly dependent on the angle of view. However, an important fraction of this radiation comes from a lim- ited solid angle and the ear was almost vertical so the sides point towards the horizons. We estimated that most of atmosphere radiation absorbed by the ear, R a , came from the lower atmosphere, which allows us to write R a ≈ σ T 4 a σ is Stephan–Boltzmann constant, equal to 5.67 × 10 − 8 W m − 2 K − 4 . Since the temper- atures of maize leaves vary linearly from the bottom to the top of vegetation, between T sol and air temper- ature T a Khabba et al., 1999b, long wave radiation balance of ear can be simplified as R lw = 1 2 σ T 4 sol − T 4 s + 1 2 σ T 4 a − T 4 s 8 where T s is the external temperature of the husk. Using this equation, the overestimation of energy by the first part of the right-hand term was almost balanced by the second part Khabba et al., 1999b. 2.3. Sensible heat exchange, H s Considering the ear as a cylinder, with a variable section, placed in air at temperature T as , the convective heat flux density, H s , can be written as H s = ρC p T s − T as r s = hT s − T as 9 where ρ is the density of air kg m − 3 , C p its spe- cific heat at constant pressure J kg − 1 K − 1 and r s the thermal diffusion resistance of husk leaves s m − 1 . The expression of convective heat-transfer coeffi- cient, h, depends on the average ear radius, r e , and on the dimensionless Nusselt number Nu Monteith and Unsworth, 1990: h = κ Nu 2r e 10 where κ is the air thermal conductivity 0.0257 W m − 1 k − 1 at 20 ◦ C. Nu can be expressed as a function of ei- ther the Reynolds number Re = u e 2r e υ, in the case of forced convection or the Grashof number Gr = 8r 3 e gβT s − T as υ in the case of free convection. To estimate the possible sizes of transfer coefficients for these two regimes, the ear was treated as a cylin- der of an average radius of 2.8 cm. Empirical relations derived from literature e.g. Kreith, 1958; Leontiev, 1979; Monteith and Unsworth, 1990; Cellier et al., 1993 gave Nu forced = 0.62Re 0.47 for forced convection 11 Nu free = 0.47Gr 0.25 for free convection 12 For wind speeds between 0.2 and 2 m s − 1 , h forced lies between 6.38 and 18.82 W m − 2 K − 1 . However, for T s − T as between 1 and 5 ◦ C and T a between 10 and 30 ◦ C, h free is expected to fall between 0.19 and 0.30 W m − 2 K − 1 . We will therefore assume that free convection was insignificant most of the time for maize ears under field conditions. This conclusion is consistent with those of Smart and Sinclair 1976 for spherical fruit and of Cellier et al. 1993 in the case of the apex of maize during early growth stages. The wind speeds at the ear level, u e , was derived from the wind speed, V s , measured at height z s using logarithmic wind profile above the canopy and an ex- ponential wind profile within the canopy Khabba et al., 1999b: u e = V s expaLz e h c − 1 logz s − dz 13 where L is the leaf area index and a an empirical co- efficient. d and z are, respectively, the zero plane dis- placement and roughness length. They were estimated from canopy height h c Armbrust and Bilbro, 1997; Kustas et al., 1989; Zhang and Gillespie, 1990; Bus- sière and Cellier, 1994; Sauer et al., 1996: z = 0.13h c 14 d = 0.67h c 15 2.4. Evaporation, λ E The latent heat flux density can be expressed as λ E = ρC p e s − e a γ r ex + r i 16 where γ is the psychrometric constant ≈66 Pa K − 1 , and e a and e s are, respectively, the vapour pressure of air and inside substomatal chambers of the leaves 136 S. Khabba et al. Agricultural and Forest Meteorology 106 2001 131–146 which constitute the husk. The external resistance to the vapour transfer, r ex , is a combination of both tur- bulent exchange resistance between canopy and bulk air aerodynamic resistance and resistance to air mass exchange within the canopy referred to as canopy re- sistance. For turbulent convection, r ex is the same as r s Jones et al., 1983; Toole and Real, 1986; Cellier et al., 1993. r i is the husk stomatal resistance and was assumed to be only due to the resistance of the stom- ata of the external face of the husk. The vapour pressures e a and e s were written as a functions of T d dew point temperature and T s , respectively: e a = P T d and e s = P T s . Using Taylor’s first-order series, the vapour pressure differ- ence in Eq. 16 can then be linearised into e s − e a = 1T s − T d 17 where 1 is the slope of saturation vapour curve. It will be estimated at air temperature T a which is al- ways included between T s and T d , and can be taken as 1 2 T s + T d . The estimation of stomatal resistance is of major concern in all the models involving relations between plant and atmosphere, and it has presently no universal solutions Tolk et al., 1995. To our knowl- edge, no investigations have been made on the stomatal resistance in the case of maize ear. We had therefore to derive ourselves a relation to calculate r i from our ex- perimental measurements in Belgium explained be- low. The model fitting method Powell, 1984; Khabba et al., 1999a was used, i.e. r i was adjusted to obtain a good fit of simulated temperatures to measured tem- peratures. The temperature at chosen points within the ear as explained below was measured as a func- tion of time. r i was estimated by minimising the sums of squares of the differences between observed and simulated temperatures least-squares method. 2.5. Energy exchange by conduction into the ear, H c For each elementary volume at husk surface, the heat flux exchanged by conduction into the ear was described by H c = − κ s gradT 18 where κ s is the thermal conductivity of husk layer and T the temperature. The thermal exchange within each one of the three layers of the ear; cob, grain or husk, was described by Fourier’s law ∂T ∂t = κ e ρ e C pe 1 r ∂ ∂r r ∂T ∂r + 1 r 2 ∂ 2 T ∂θ 2 + ∂ 2 T ∂z 2 19 where t is the time, κ e , ρ e and C p e are the thermal conductivity, density and heat capacity of each layer of the ear. Eq. 19 take into account heat transfer in the radial r, polar θ and longitudinal z direction. At the interface between joining layers, husk–grain, husk–cob or grain–cob, heat conduction was assumed purely conductive: T X−ε2 = T X+ε2 20 κ e gradT | X−ε2 = κ e gradT | X+ε2 21 where X is the interface level and ε a very small real number compared with step interval 1r and 1z ε = 0.1 mm. Eqs. 19–21 with appropriate external boundary conditions Eqs. 1, 8, 9 and 16 were solved us- ing a finite difference scheme and alternating direc- tion method Samorsky and Gulin, 1973. A mesh of r, θ, z = 27, 36, 41 was found to be sufficient to model the problem accurately. 2.6. Procedure of model testing Eq. 13 was adjusted by Khabba et al. 1999b using the wind speed measurements as those made in Belgium described below. The value obtained for the empirical coefficient a was 0.51 r 2 = 0.96. In the absence of an estimate of stomatal resistance, the test of the model was made as follows: • Before estimating ear stomatal resistance with the model, the equations used to calculate ear radiation balance were tested. This was done by comparing measured and calculated downward and upward ra- diation received at the ear position. The flux of ra- diation was calculated in the model by R bΩ s + R dd + R sr + 1 2 σ T 4 sol + T 4 a 22 • The stomatal resistance r i was estimated using the model method described above. The values obtained were used to determine a relationship between r i and micrometeorological factors. The relationship obtained was inserted in the model. S. Khabba et al. Agricultural and Forest Meteorology 106 2001 131–146 137 • Finally the model was tested using experimental observations made in Morocco June 1997 and in Belgium September 1998.

3. Experiment