## Appendix A. Supplementary Description of the Variable Formula in the Crop Module

1. The Calculation of Stage Development Variables

(1) Thermal Effect

The

TE can be described by the sinusoidal exponential equation, as shown below [

27]:

where

TE_{i} is the hour thermal effect;

DTE is the daily thermal effect;

T_{i} is the hour temperature of air, °C,

i = 1, 2, …, 24;

T_{b} is the base temperature of crop growth, °C;

T_{0} is the optimum temperature of crop growth, °C;

T_{m} is the maximum temperature of crop growth, °C;

ts is temperature sensitivity, which is genetically determined by genetic parameters. For winter wheat, the values of

T_{b},

T_{0}, and

T_{m} are shown in

Table A1.

**Table A1.**
The parameters of thermal effect for winter wheat.

**Table A1.**
The parameters of thermal effect for winter wheat.

| T_{b} (°C) | T_{0} (°C) | T_{m} (°C) |
---|

Emergence date to double ridge date | 0 | 20 | 32 |

Double ridge date to heading date | 3.3 | 22 | 32 |

Heading date to maturity | 8 | 25 | 35 |

(2) Vernalization Effect

Vernalization is the induction of the crop flowering process by exposure to the prolonged low temperature [

48], which can also prevent the damage of the cold-sensitive flowering meristem during the winter [

49]. The calculation of the

VE in wheat is:

where

VE_{i} is the hour vernalization effect;

T_{bv} is the minimum vernalization temperature, −1 °C;

T_{ol} is the lower limit value of the optimum vernalization temperature range, 1 °C;

T_{ou} is the upper limit value of the optimum vernalization temperature range, °C;

T_{mv} is the highest vernalization temperature, °C;

vef is the vernalization effect factor.

T_{ou},

T_{mv}, and

vef can be calculated as:

where

PVT is the physiological vernalization time, the value shown in

Appendix C. When the temperature is higher than 27 °C, and the number of vernalization days does not exceed 1/3 of the physiological vernalization time, the wheat will devernalize. The devernalization effect (

VED) increases with increasing temperature, which can be written as:

Daily vernalization effect (

DVE) is affected by the effects of

VE, and

VED:

Therefore, the vernalization days (

VD) on

t day can be expressed as:

The vernalization process (

VP) is expressed by the ratio of

VD to

PVT. When

VD is equal to

PVT or

VP reaches 1, the vernalization is considered to be over:

(3) Photoperiod Effect

The

PE is the sensitivity of stage developmental rate to photoperiod or daytime, mainly related to crop varieties and the length of theoretical sunshine hours and can be described by [

50]:

where

PS is the photoperiod sensitivity (value shown in Appesdix C); and

DL is the theoretical sunshine duration, h.

2. The calculation of absorbed light

The calculation of

I_{L} (photosynthetically active radiation intensity at

L depth of crop canopy) in Equation (5) was based on Cao et al. [

27], which is described by several formulas below:

where

I_{0L} is the photosynthetically active radiation from canopy top to depth

L, J·m

^{−2}·s

^{−1};

LAI(L) is the accumulated leaf area index from canopy top to depth

L;

κ is the extinction coefficient;

ρ is the canopy emissivity;

σ is the single leaf dissipation coefficient;

β is the solar elevation, rad;

t_{h} is the apparent solar time, h;

PARCAN is the photosynthetically active radiation at canopy top, J·m

^{−2}·s

^{−1};

n is the actual number of sunshine hours, h;

DL is the day length, h;

PAR is the light and effective radiation on the upper bound of the atmosphere, J·m

^{−2}·s

^{−1};

SC is the solar constant value, 1395 J·m

^{−2}·s

^{−1};

J is the date ordinal in year;

RDN is the solar constant fraction at a certain date (

J) and a certain latitude (

ϕ);

ϕ is the latitude, rad;

δ is the solar declination, rad.

3. The Calculation of the Influence Factor

The calculation of the influence factors in Equation (5) is depicted as below [

27]:

(1) CO

_{2} concentration influence factor

where

C_{x} is the CO

_{2} concentration, ppm;

C_{0} is the reference CO

_{2} concentration (usually 340 ppm);

α is the empirical coefficient—for winter wheat it is 0.8. In this text there is no CO

_{2} input, and the

FC is 0.95.

(2) Temperature Influence Factor

where

T_{mean} is the daily mean temperature, °C.

(3) Water Influence Factor

where

θ is the average water content in 0–30 cm soil layer;

θ_{OL} is the lower limit of optimum soil water content;

θ_{OH} is upper limit of optimum soil water content;

θ_{WP} is the wilting point soil content. Values are shown in

Table A2.

(4) Nitrogen Influence Factor

where

N is the plant actual nitrogen content;

NL is the plant maximum nitrogen content; and

NC is the plant minimum nitrogen content.

4. The Calculation of Respiration

The calculation of respiration is developed from Cao et al. [

27]. The respiration includes maintain respiration (

RM), growth respiration (

RG), and photorespiration (

RP). Plants continuously provide energy to the organism by

RM to keep its existing biochemical and physiological state, which is sensitive to temperature and in proportion to the amount of assimilation:

where

T_{0} is the optimum temperature of respiration, °C;

Q_{10} is the temperature coefficient of respiration, °C;

T_{mean} is the daily mean air temperature, °C;

RM (

T_{0}) is the sustained respiration coefficient at

T_{0}. Values are shown in

Table A2.

RG is insensitive to temperature and depends mainly on the rate of photosynthesis:

where

Rg is growth respiration coefficient, gCO

_{2}·gCO

_{2}^{−1}.

RP is related to the amount of assimilation. In C3 crop the

RP increases with increasing temperature and increasing light intensity while in C4 crops is almost completely suppressed and negligible:

where

Rp(

T_{0}) is the photorespiration coefficient at

T_{0}, gCO

_{2}·gCO

_{2}^{−1}; and

T_{day} is the daytime temperature, °C.

## Appendix B. Supplementary Description of the Variable Formula in the SPAC Module

1. The Calculation of Radiation

(1) The total short-wave radiation

R_{g} is calculated from [

34]:

where

G is the daily total shortwave radiation, W·m

^{−2};

SN is the time of solar noon;

N is the theoretical sunshine hours in a day;

α_{0} is the sunrise sunset angle;

L_{m} is the local longitude;

dN is the time difference;

dn is the daily ordinal number;

a_{0} to

a_{4} are Fourier expansion coefficients, values 0.0075, 0.001868, 0.0032077, −0.012615, and −0.04089, respectively.

where

a,

b are empirical coefficients, with values of 0.105 of 0.708 (refer to Sun [

51]);

n is the actual number of daylight hours;

G_{m} is the total shortwave radiation without atmospheric weakening;

η is the ratio of the solar and land distance to the average distance;

ψ is the local latitude; and

δ is the declination of the day.

(2) The effective longwave radiation

F can be calculated from an empirical formula [

34], which adopts the Stefan-Boltzmann Law:

where

σ is the Stefan-Boltzmann constant, 5.673 × 10

^{−8} Wm

^{−2}·K

^{−4};

μ is the radiation ratio, with the value shown in

Table A2;

T_{a} is the atmospheric temperature, °C;

T_{s} is the underlying surface temperature, °C; and

H_{a} is the absolute humidity of air.

2. The Calculation of the Sensible Heat and Latent Heat

According to the theories of aerodynamics and micro meteorology,

H,

H_{v},

H_{s},

LE,

LE_{v}, and

LE_{s} can be calculated by [

36]:

where

ρ,

C_{p}, and

γ are air density, kg·m

^{−3}, constant pressure specific heat capacity, 1008.3 J·kg

^{−1}·K

^{−1}, and hygrometer constant, hPa·K

^{−1}, respectively;

e_{a} and

T_{a} are the vapour pressure (hPa) and temperature (°C) of the air at the reference altitude;

e_{b} and

T_{b} are the vapour pressure and temperature of the canopy air;

T_{v} and

e_{v} are the leaf temperature and the vapour pressure in the interstitial space of the leaf stomata and mesophyll cells;

T_{1} and

e_{1} are the temperature and vapour pressure of the soil surface;

r_{ba},

r_{vb},

r_{sb},

r_{v}, and

r_{s} are the aerodynamic resistance of the hydrothermal transmission from the canopy to the atmosphere, s·m

^{−1}, the canopy boundary layer resistance of the hydrothermal transmission from the leaf to canopy, s·m

^{−1}, the aerodynamic resistance of the hydrothermal transmission from the soil surface to the canopy, s·m

^{−1}, the canopy total stomatal resistance, s·m

^{−1}, and the soil surface evaporation resistance, s·m

^{−1}, respectively. Each resistance needs to be determined by theoretical or empirical methods.

3. The Calculation of Water–Heat Transfer Resistance

(1) Aerodynamic resistance of the water heat travels from the canopy to the atmosphere [

52]:

where

u is the wind speed at 2 m, m·s

^{−1};

κ is the Karman constant, 0.4;

d is the zero plane displacement, m;

z_{0} is the surface roughness of canopy, m. For winter wheat,

d = 0.56 h,

z_{0} = 0.3(h − d) = 0.132 h, where

h is the plant height.

(2) The canopy boundary layer resistance of the water heat travels from the leaf to the canopy [

36]:

where

a is the attenuation coefficient of wind speed in the canopy, value is 3;

b = 0.01 m·s

^{−0.5};

w is the leaf width, m;

u_{top} is the canopy top wind speed, m·s

^{−1}.

(3) Aerodynamic resistance of the water heat travels from soil surface to canopy layer [

36]:

where

K_{d}(

h) is the momentum vortex diffusion rate of the canopy top, m

^{2}·s

^{−1};

α is the attenuation coefficient, −2;

z_{0}’ is the soil surface roughness, m, 0.01 m.

(4) Stomatal resistance

According to the experiment data, Wu [

52] generalize the diurnal variation of stomatal resistance under the condition of irrigation to a linear function, which is described as:

where

t is the time of day;

t_{r} and

t_{s} are the times of sunrise and sunset; and

r_{v,max} and

r_{v,min} are related to the leaf area index:

(5) Soil evaporation resistance

An empirical formula based on experimental data derived by Lin [

53] is used in this paper:

where

θ_{s} is the saturated water content of the soil;

θ is the average soil moisture content of the surface at 5 cm;

a, b_{1}, and

b_{2} are empirical constants, 5, 33.5, and 2.3, respectively.