# A Coupled Model for Simulating Water and Heat Transfer in Soil-Plant-Atmosphere Continuum with Crop Growth

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}for W0, W1, W2, W3, and W4 is 0.90, 0.88, 0.90, 0.91, and 0.79, and the root mean square error (RMSE) is 17.24 mm, 27.65 mm, 20.47 mm, 22.35 mm, and 12.88 mm, respectively. For soil temperature along the soil profile, the R

^{2}ranges between 0.96 and 0.98, and the RMSE between 1.22 °C and 1.94 °C. For LAI, the R

^{2}varied from 0.76 to 0.96, and the RMSE from 0.52 to 0.67. We further compared the simulation results by CropSPAC and its two detached modules, i.e., crop and the SPAC modules. Results demonstrate that the coupled model could better reflect the interactions between crop growth and soil moisture condition, more suitable to be used under deficit irrigation conditions.

## 1. Introduction

## 2. Model Conceptualization and Formulation

#### 2.1. Coupling of Crop Growth and SPAC Water–Heat Transport

#### 2.2. Crop Module

#### 2.2.1. Simulation of Crop Stage Development

_{t}is the physiological development time without consideration of crop variety impact, which was accumulated by the daily physiological effect (DPE):

#### 2.2.2. Simulation of Biomass Accumulation

_{2}·ha

^{−1}·h

^{−1}; PLMX

_{0}is the maximum photosynthetic rate of leaves, kgCO

_{2}·ha

^{−1}·h

^{−1}; ε is the initial utilization efficiency of absorbed light, kgCO

_{2}·ha

^{−1}·h

^{−1}·(J·m

^{−2}·s

^{−1})

^{−1}; I

_{L}is the photosynthetically active radiation intensity at L (depth of crop canopy), J·m

^{−2}·s

^{−1}; FH is the water influence factor; FN is the nitrogen influence factor (note that the current CropSPAC model could not simulate the soil nitrogen migration, however, any nitrogen stress could be empirically indicated by this parameter); FT is the temperature influence factor; FC is the CO

_{2}concentration influence factor. Calculation methods and details are shown in Appendix A.

_{2}·hm

^{−2}·day

^{−1}; RM is consumption of maintain respiration, kgCO

_{2}·hm

^{−2}·day

^{−1}; RG is consumption of growth respiration, kgCO

_{2}·hm

^{−2}·day

^{−1}; RP is consumption of photorespiration, kgCO

_{2}·hm

^{−2}·day

^{−1}, with details shown in Appendix A.

^{−2}·day

^{−1}; ξ is conversion coefficients between CO

_{2}and organic compounds.

_{DAY}is accumulation of dry biomass, kgDM·hm

^{−2}·day

^{−1}.

#### 2.2.3. Dry Biomass Partitioning and Yield

_{TOP}is the aboveground part distribution index, P

_{ROOT}is the underground part distribution index, P

_{LEAF}is the leaf distribution index, P

_{EAR}is the ear distribution index, P

_{STEM}is the stem and sheath distribution index, and HI is the harvest index.

_{EAR}) was established by Cong [29]:

_{LEAF}) was given as:

^{−1}[32], and in this study it is calibrated by field data (value shown in Appendix A).

#### 2.3. SPAC Module

#### 2.3.1. Calculation of Net Radiation and Water-Heat Transport in Crop Canopy

_{n}is the net radiation, W·m

^{−2}; R

_{g}is total short-wave radiation, W·m

^{−2}; A is the land surface albedo; F is net long-wave radiation, W·m

^{−2}. The R

_{g}, and F can be calculated by theoretical or empirical methods shown in the Appendix B [34], the value of A can be seen in Appendix C, Table A2.

_{n}can be divided into two parts:

_{v}is the net radiation absorbed by crop canopy, W·m

^{−2}; R

_{s}is the net radiation absorbed by ground surface, W·m

^{−2}.

_{s}is calculated from the empirical formula summarized from field observations [35]:

_{v}is the latent heat flux by leaf transpiration; H

_{v}is the sensible heat flux between crop leaves and canopy air; LE

_{s}is the latent heat flux by soil surface evaporation; H

_{s}is the sensible heat flux between soil and canopy air; G is the downward heat flux in the soil surface; H is the sensible heat flux between canopy air and the above atmosphere (usually considered at the reference height); LE is the latent heat flux between canopy air and the reference height atmosphere. The unit of the above variables is W·m

^{−2}. Schematic diagrams of energy distribution between atmospheric-ground interfaces and resistance of the hydrothermal transmission are shown in Figure 5. Detailed calculation methods are shown in Appendix B.

#### 2.3.2. Calculation of Crop Transpiration under Water Stress in Root Zone

_{v}), which induce the reduction of canopy transpiration. Plant physiologists have studied the effects of soil moisture changes on leaf water potential and stomatal resistance [37], but due to the complex mechanism involved, it is difficult to quantify directly the solve leaf water potential and stomatal resistance under soil water stress. Neglecting the water storage variation in the crop, we assume the root water uptake equals the canopy transpiration. Therefore, the response of canopy transpiration to soil water stress can be reflected by the actual root water uptake described by the soil water stress coefficient of root water uptake k(ψ). The actual transpiration of crops under the condition of water stress, E

_{v}, can be calculated by:

_{vp}is the potential transpiration without water stress in root zone.

_{v}, see Equation (A42)) can be replaced by Equation (26):

_{v}

_{0}is the canopy total stomatal resistance under no soil water stress, s·m

^{−1}.

#### 2.3.3. Water and Heat Transfer in Soil with Root Water Uptake

^{3}·cm

^{−3}; T is the soil temperature, °C; t is the time, s; z is the depth from the surface, m; D

_{w}is the coefficient of soil water diffusion, m

^{2}·s

^{−1}; D

_{wh}are the diffusion coefficients of temperature gradients to water flow, m

^{2}·s

^{−1}·°C

^{−1}; K

_{w}is soil water conductivity, m·s

^{−1}; C

_{v}is soil volumetric heat capacity, J·m

^{−2}·°C

^{−1}; K

_{h}is the soil thermal conductivity, J· s

^{−1}·°C

^{−1}; s(z,t) is the distribution function of the actual water uptake rate of roots, m

^{3}·m

^{−3}·s

^{−1}; S

_{h}is the heat source sink term, J·m

^{−2}·°C

^{−1}, generally negligible.

_{vp}, which is not limited by soil water stress, equal to the potential total root water uptake rate, i.e., the integration of potential root water uptake rate along the entire root zone considering the root density distribution. The actual root water uptake rate s(z, t) is equal to the potential root water uptake rate s

_{0}(z, t) multiplied by water stress response function α (ψ, π) [39]:

_{50}is the specific soil water potential, when E

_{vp}decreases by 50%(ψ

_{50}≈ −0.43 MPa); Z

_{r}is the depth of the root zone.

## 3. Experiment and Model Input

#### 3.1. Study Area and Experiment

^{−3}, and the depth of water table is mostly larger than 5 m in this area.

^{2}(5 m × 10 m), as shown in Figure 6. It includes six rows for five irrigation treatments (two rows for W2 treatment), and four columns for three fertilization treatments (two columns for medium level fertilization treatment). Since only column FM-1 has crop growth index monitoring, the medium fertilization with five levels of water treatments were used for model calibration and validation. In those treatments, the level of fertilization is the same, which was reflected in the factor FN (e.g., FN is assumed to be 0.95) in CropSPAC. The details of fertilization volume, irrigation volume, and time were shown in Table 1.

#### 3.2. Model Input

_{h}and C

_{v}, mainly vary with the soil water content, can be described by Johansen model [44].

## 4. Simulation Results

#### 4.1. Soil Water Content

^{2}in W0, W2, W4, W1, and W3 is 0.90, 0.90, 0.79, 0.88, and 0.91, and is RMSE 17.24 mm, 20.47 mm, 12.88 mm, 27.65 mm, and 22.35 mm, respectively.

^{2}on March 25, April 27, May 20, and June 2 in W4 of 0.92, 0.24, 0.74, 0.61, March 25, April 27, May 20, and June 5 in W1 0.89, 0.78, 0.23, 0.91, and the corresponding RMSE of 0.01, 0.03, 0.02, 0.03, 0.01, 0.05, 0.03, and 0.02, respectively.

#### 4.2. Soil Temperature

^{2}varied between 0.96 and 0.98, and RMSE ranges from 1.22 °C to 1.94 °C. However, it seems the simulated results systematically underestimated soil temperature generally in the early crop stage. The reason may be, (1) according to the observed air temperature, there is a large drop on March 22. It is captured in the simulation with the soil temperature dropped accordingly, especially in the near surface layers. However, strangely the observed data did not show this drop, possibly because some artificial measure had been taken to keep the soil temperature which was unable to be captured in our model; and (2) due to the uncertainty of local crop parameters, the simulated LAI tended to be higher in the early crop stage. Thus, the larger canopy meant larger interception of radiation and less radiation absorbed by the soil surface. Therefore, the simulated soil temperature tended to be small in the observed data in the early crop stage. With more work on the accuracy of crop data and the detailed investigation of agriculture management, the simulation performance was able to be improved in future research.

#### 4.3. Leaf Area Index

^{2}ranging between 0.76 to 0.96, RMSE between 0.52 and 0.67. From early March when winter wheat turned green, leaf area and dry biomass gradually increased with the increase of air temperature. When winter wheat entered the grain filling stage, the LAI reached the maximum value. Then the leaves got senescent and LAI declined, and the dry biomass accumulated by the plant transfers from the organs to the ear. The peak and time when the LAI reached this maximum value in the three treatments were different as shown in Figure 11. For example, in W4 treatment with the adequate irrigation water supply, it is in the beginning of May that LAI reached its peak value of 5.6, while in W2 and W0 treatments the time is in early April, with the peak values also smaller at between 4 and 5. This illustrated that deficit irrigation limited the growth of crop and the extension of the foliage, and led to the crop early mature. This is also a kind of physiological protection measures for plants in the shortage of water supply, mature in advance to avoid large-scale production loss, as demonstrated by other research [46,47].

#### 4.4. Biomass and Yield

^{2}varied from 0.93 to 0.99 and RMSE from 993 kg/ha to 1729 kg/ha in different treatments. The increase in irrigation contributed to the accumulation of dry biomass. The reduction in water directly led to a reduction in dry biomass accumulation and then affected the final yield. As a whole, the simulation of the CropSPAC on the accumulation of winter wheat biomass and the final yield is reliable.

## 5. Discussion on the Coupling Effect of Model

#### 5.1. Comparison between CropSPAC and the Detached Crop Module

#### 5.2. Comparison between CropSPAC and SPAC

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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

_{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.

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 |

_{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:

_{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:

_{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.

_{2}concentration influence factor

_{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.

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

_{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.

_{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.

_{2}·gCO

_{2}

^{−1}.

_{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

_{g}is calculated from [34]:

^{−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.

_{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.

^{−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.

_{v}, H

_{s}, LE, LE

_{v}, and LE

_{s}can be calculated by [36]:

_{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.

^{−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.

^{−0.5}; w is the leaf width, m; u

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

^{−1}.

_{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.

_{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:

_{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.

## Appendix C. Parameters of the CropSPAC

Sort | Symbol | Parameter Name | Unit | Value | Source |
---|---|---|---|---|---|

Genetic parameter | ts | Temperature sensitivity | 0.9 | ||

PVT | Physiological vernalization time | day | 50 | ||

PS | Photoperiod sensitivity | 0.005 | [27] | ||

BDF | Basic developmental factor | 0.99 | |||

HI | Harvest index | 0.5 | |||

SLA | leaf weight per unit | ha/kg | 0.0022 | ||

Parameters in water influence factor | θ_{OH} | Upper limit of optimum soil water content | cm^{3}/cm^{3} | 0.35 | Calibrated with the observed data |

θ_{OL} | Lower limit of optimum soil water content | cm^{3}/cm^{3} | 0.18 | ||

θ_{WP} | Wilting point soil content | cm^{3}/cm^{3} | 0.05 | ||

Influence factor | FC | CO_{2} concentration influence factor | 0.95 | [27] | |

FN | Nitrogen influence factor | 0.95 | |||

Photosynthesis | κ | Extinction coefficient | 0.6 | [27] | |

PLMX_{0} | Maximum photosynthetic rate of leaves | kgCO_{2}·ha^{−1}·h^{−1} | 40 | ||

ε | Initial utilization efficiency of absorbed light | kgCO_{2}·ha^{−1}·h^{−1}/(J·m^{−2}·s^{−1}) | 0.5 | ||

σ | Single leaf dissipation coefficient | 0.2 | |||

Respiration | T_{0} | Optimum temperature of respiration | °C | 25 | [27] |

Q_{10} | Temperature coefficient of respiration | 2 | [27] | ||

RM(T_{0}) | Sustained respiration coefficient at T_{0} | gCO_{2}/gCO_{2} | 0.010 | Calibrated with the observed data | |

Rg | Growth respiration coefficient | gCO_{2}/gCO_{2} | 0.26 | ||

Rp(T_{0}) | Photorespiration coefficient at T_{0} | gCO_{2}/gCO_{2} | 0.20 | ||

Radiation | A | Surface albedo | 0.25 | [34] | |

μ | Radiation ratio | 0.97 | [34] | ||

A | Net radiation distribution coefficient | 0.3973 | [35] | ||

B | Net radiation distribution coefficient | 1.036 | [35] |

## References

- Passioura, J.B. Plant–Water Relations. Encyclopedia of Life Sciences (ELS); John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2010; pp. 1–7. [Google Scholar]
- Nilson, S.E.; Assmann, S.M. The control of transpiration. Insights from Arabidopsis. Plant Physiol.
**2007**, 143, 19–27. [Google Scholar] [CrossRef] [PubMed] - Liu, B.; Yang, L.; Zeng, Y.; Yang, F.; Yang, Y.; Lu, Y. Secondary crops and non-crop habitats within landscapes enhance the abundance and diversity of generalist predators. Agric. Ecosyst. Environ.
**2018**, 258, 30–39. [Google Scholar] [CrossRef] - Silva, V.D.P.R.D.; Silva, R.A.E.; Maciel, G.F.; Braga, C.C.; Junior, J.L.C.D.S.; Souza, E.P.D.; Almeida, R.S.R.; Silva, A.M.T.; Holanda, R.M.D. Calibration and validation of the AquaCrop model for the soybean crop grown under different levels of irrigation in the Motopiba region, Brazil. Cienc. Rural
**2018**, 48, 1–8. [Google Scholar] [CrossRef] - Goyne, P.J.; Meinke, H.; Milroy, S.P.; Hammer, G.L.; Hare, J.M. Development and use of a barley crop simulation model to evaluate production management strategies in north-eastern Australia. Aust. J. Agric. Res.
**1996**, 47, 1470–1477. [Google Scholar] [CrossRef] - Badenko, V.L.; Garmanov, V.V.; Ivanov, D.A.; Savchenko, A.N.; Topaj, A.G. Prospects of application of dynamic crop models in the problems of midterm and long-term planning of agricultural production and land management. Russ. Agric. Sci.
**2015**, 41, 187–190. [Google Scholar] [CrossRef] - Wang, L.; Agyemang, S.A.; Amini, H.; Shahbazi, A. Mathematical modeling of production and biorefinery of energy crops. Renew. Sustain. Energy Rev.
**2015**, 43, 530–544. [Google Scholar] [CrossRef] - Lizaso, J.I.; Boote, K.J.; Jones, J.W.; Porter, C.H.; Echatte, L.; Westgate, M.E.; Sonohat, G. CSM-IXIM: A new maize simulation model for DSSAT version 4.5. Agron. J.
**2011**, 103, 766–779. [Google Scholar] [CrossRef] - Liu, B.; Asseng, S.; Liu, L.; Tang, L.; Cao, W.; Zhu, Y. Testing the responses of four wheat crop models to heat stress at anthesis and grain filling. Glob. Chang. Biol.
**2016**, 22, 1890–1903. [Google Scholar] [CrossRef] [PubMed] - Maiorano, A.; Martre, P.; Asseng, S.; Ewert, F.; Muller, C.; Rotter, R.P.; Ruane, A.C.; Semenov, M.A.; Wallach, D.; Wang, E.; et al. Crop model improvement reduces the uncertainty of the response to temperature of multi-model ensembles. Field Crops Res.
**2017**, 202, 5–20. [Google Scholar] [CrossRef] - Williams, J.R.; Jones, C.A.; Kiniry, J.R.; Spanel, D.A. The EPIC crop growth model. Trans. ASABE
**1989**, 32, 497–511. [Google Scholar] [CrossRef] - Diepen, C.A.; Wolf, J.; Keulen, H.; Rappoldt, C. WOFOST: A simulation model of crop production. Soil Use Manag.
**1989**, 5, 16–24. [Google Scholar] [CrossRef] - Pathak, H.; Li, C.; Wassmann, R. Greenhouse gas emissions from Indian rice fields: Calibration and upscaling using the DNDC model. Biogeosciences
**2005**, 2, 77–102. [Google Scholar] [CrossRef] - Jones, J.W.; Hoogenboom, G.; Porter, C.H.; Boote, K.J.; Batchelor, W.D.; Hunt, L.A.; Wilkens, P.W.; Singh, U.; Gijsman, A.J.; Ritchie, J.T. The DSSAT cropping system model. Eur. J. Agron.
**2003**, 18, 235–265. [Google Scholar] [CrossRef] - Raes, D.; Steduto, P.; Hsiao, T.C.; Fereres, E. AquacropThe FAO crop model to simulate yield response to water: II. main algorithms and software description. Agron. J.
**2009**, 101, 438–447. [Google Scholar] [CrossRef] - Yang, J.; Mao, X.; Wang, K.; Yang, W. The coupled impact of plastic film mulching and deficit irrigation on soil water/heat transfer and water use efficiency of spring wheat in Northwest China. Agric. Water Manag.
**2018**, 201, 232–245. [Google Scholar] [CrossRef] - Philip, J.R. Plant water relations: Some physical aspects. Annu. Rev. Plant Physiol.
**1966**, 17, 245–268. [Google Scholar] [CrossRef] - Yang, Y.; Shang, S.; Guan, H. Development of a soil-plant-atmosphere continuum model (HDS-SPAC) based on hybrid dual-source approach and its verification in wheat field. Sci. China Technol. Sci.
**2012**, 55, 2671–2685. [Google Scholar] [CrossRef] - Deng, Z.; Guan, H.; Hutson, J.; Forster, M.A.; Wang, Y.Q.; Simmons, C.T. A vegetation-focused soil-plant-atmospheric-continuum model to study hydrodynamic soil-plant water relations. Water Resour. Res.
**2017**, 53, 4965–4983. [Google Scholar] [CrossRef] - Federer, C.A. A soil-plant-atmosphere model for transpiration and availability of soil water. Water Resour. Res.
**1979**, 15, 555–562. [Google Scholar] [CrossRef] - Gou, S.; Miller, G. A groundwater–soil–plant–atmosphere continuum approach for modelling water stress, uptake, and hydraulic redistribution in phreatophytic vegetation. Ecohydrology
**2014**, 7, 1029–1041. [Google Scholar] [CrossRef] - Zhang, X.; Xiao, Y.; Wan, H.; Deng, Z.; Pan, G.; Xia, J. Using stable hydrogen and oxygen isotopes to study water movement in soil-plant-atmosphere continuum at Poyang Lake wetland, China. Wetl. Ecol. Manag.
**2017**, 25, 1–14. [Google Scholar] [CrossRef] - Kroes, J.G.; Huygen, J.; Vervoort, R.W. User’s Guide of SWAP Version 2.0; Simulation of Water Flow, Solute Transport and Plant Growth in the Soil-Water-Atmosphere-Plant Environment; DLO Winand Staring Centre: Wageningen, The Netherlands, 1999. [Google Scholar]
- Arnold, J.G.; Srinivasan, R.; Muttiah, S.; Williams, J.R. Large area hydrologic modeling and assessment part I: Model development. J. Am. Water Resour. Assoc.
**1998**, 34, 73–89. [Google Scholar] [CrossRef] - Chen, F.; Mitchell, K.; Schaake, J.; Xue, Y.; Pan, H.; Koren, V.; Duan, Q.Y.; Ek, M.; Betts, A. Modeling of land surface evaporation by four schemes and comparison with FIFE observations. J. Geophys. Res. Atmos.
**1996**, 101, 7251–7268. [Google Scholar] [CrossRef] - Ek, M.; Mitchell, K.; Lin, Y.; Rogers, E.; Grunmann, P.; Koren, V.; Gayno, G.; Tarpley, J. Implementation of Noah land surface model advances in the national centers for environmental prediction operational mesoscale Eta model. J. Geophys. Res. Atmos.
**2003**, 108, 8851. [Google Scholar] [CrossRef] - Cao, W.; Luo, W. Crop System Simulation and Intelligent Management; Higher Education Press: Beijing, China, 2003; pp. 27–33, 57–63. ISBN 7-04-011756-8. [Google Scholar]
- Cao, W.; Moss, D.N. Modelling phasic development in wheat: A conceptual integration of physiological components. J. Agric. Sci.
**1997**, 129, 163–172. [Google Scholar] [CrossRef] - Cong, Z.; Lei, Z.; Hu, H.; Yang, S. Study on coupling between winter wheat growth and water-heat transfer in soil-plant-atmosphere continuum I. Model. J. Hydraul. Eng.
**2005**, 36, 575–580. [Google Scholar] [CrossRef] - Goudriaan, J. A simple and fast numerical method for the computation of daily totals of crop photosynthesis. Agric. For. Meteorol.
**1986**, 38, 249–254. [Google Scholar] [CrossRef] - Liu, T.; Cao, W.; Luo, W.; Wang, S.; Guo, W.; Zou, W.; Zhou, Q. Quantitative simulation on dry matter partitioning dynamic in wheat organs. J. Triticeae Crops.
**2001**, 21, 25–31. [Google Scholar] [CrossRef] - Liu, T.; Cao, W.; Luo, W.; Guo, W. Simulation on leaf area index in wheat. J. Triticeae Crops
**2001**, 21, 25–31. [Google Scholar] [CrossRef] - Mao, X.; Shang, S.; Lei, Z.; Yang, S. Study on evapotranspiration of winter wheat using SPAC model. J. Hydraul. Eng.
**2001**, 8, 7–11. [Google Scholar] [CrossRef] - Bavel, C.H.M.V.; Hillel, D.I. Calculating potential and actual evaporation from a bare soil surface by simulation of concurrent flow of water and heat. Agric. Meteorol.
**1976**, 17, 453–476. [Google Scholar] [CrossRef] - Kang, S.; Liu, X.; Xiong, Y. Theory of Water Transport in Soil-Plant-Atmosphere Continuum and its Applicatition; Water Resources and Electric Power Press: Beijing, China, 1994; pp. 30–37. ISBN 7-120-01718-7. [Google Scholar]
- Choudhury, B.J.; Monteith, J.L. A four-layer model for the heat budget of homogeneous land surfaces. Q. J. R. Meteorol. Soc.
**1988**, 114, 373–398. [Google Scholar] [CrossRef] - Hanan, N.P.; Prince, S.D. Stomatal conductance of west-central supersite vegetation in HAPEX-Sahel: Measurements and empirical models. J. Hydrol.
**1997**, 188, 536–562. [Google Scholar] [CrossRef] - Lei, Z.; Yang, S.; Xie, S. Soil Water Dynamics; Tsinghua University Press: Beijing, China, 1999; pp. 208–214. ISBN 7-302-00208-8. [Google Scholar]
- Xu, D.; Cai, L. Simulation of infiltration and recharge based on field water balance in cropped soil. J. Hydraul. Eng.
**1997**, 12, 64–71. [Google Scholar] [CrossRef] - Yao, K.M.; Jian, W.M.; Zheng, H.S. Experimental Research Methods for Agrometeorology; Meteorological Press: Beijing, China, 1995; pp. 214–216. ISBN 7-5029-1810-8.
- Flerchinger, G.N.; Saxton, K.E. Simultaneous heat and water model of a freezing snow-residue-soil system I. Theory and development. Trans. ASAE
**1989**, 32, 0565–0571. [Google Scholar] [CrossRef] - Mualem, Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res.
**1976**, 12, 513–522. [Google Scholar] [CrossRef] - Van Genuchten, T.M. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef] - Johansen, O. Thermal Conductivity of Soils. Ph.D. Thesis, Trondheim University, Trondheim, Norway, 1975. [Google Scholar]
- Shang, S.; Li, X.; Mao, X.; Lei, Z. Simulation of water dynamics and irrigation scheduling for winter wheat and maize in seasonal frost areas. Agric. Water Manag.
**2004**, 68, 117–133. [Google Scholar] [CrossRef] - Korres, N.E.; Norsworthy, J.K.; Burgos, N.R.; Oosterhuis, D.M. Temperature and drought impacts on rice production: An agronomic perspective regarding short-and long-term adaptation measures. Water Res. Rural Dev.
**2016**, 9, 12–27. [Google Scholar] [CrossRef] - Shavrukov, Y.; Kurishbayev, A.; Jatayev, S.; Shvidchenko, V.; Zotova, L.; Koekemoer, F.; Groot, S.D.; Soole, K.; Langridge, P. Early flowering as a drought escape mechanism in plants: How can it aid wheat production? Front. Plant Sci.
**2017**, 8, 1–8. [Google Scholar] [CrossRef] - Yan, L.; Loukoianov, A.; Blechl, A.; Tranquilli, G.; Ramakrishna, W.; Sanmiguel, P.; Bennetzen, J.; Echenique, V.; Dubcovsky, J. The wheat VRN2 gene is a flowering repressor downregulated by vernalization. Science
**2004**, 303, 1640–1644. [Google Scholar] [CrossRef] [PubMed] - Yan, L.; Loukoianov, A.; Tranquilli, G.; Helguera, M.; Fahima, T.; Dubcovsky, J. Positional cloning of the wheat vernalization gene VRN1. Proc. Natl. Acad. Sci. USA
**2003**, 100, 6263–6268. [Google Scholar] [CrossRef] [PubMed] - Cao, W.; Jiang, H. Modeling thermal-photo response and development progress in wheat. J. Nanjing Agric. Univ.
**1996**, 19, 9–16. [Google Scholar] [CrossRef] - Sun, Z.; Shi, J.; Weng, D. A further research on the climatological calculation method of the global solar radiation over China. J. Nanjing Inst. Meteorol.
**1992**, 2, 21–29. [Google Scholar] [CrossRef] - Wu, Q. Numerical Simulation of Hydrothermal Migration under Field Evapotranspiration. Ph.D. Thesis, Tsinghua University, Beijing, China, 1993. [Google Scholar]
- Lin, J.; Sun, S. A study of moisture and heat transport in soil and the effect of surface resistance to evaporation. J. Hydraul. Eng.
**1983**, 7, 3–10. [Google Scholar] [CrossRef]

**Figure 1.**Coupling structure of the CropSPAC model. θ is the soil water content; Zr is the root distribution; and h is the plant height.

**Figure 2.**Calculation flowchart of the CropSPAC model. The GAI is the green area index; E

_{vp}is the potential transpiration; E

_{s}is the soil evaporation rate; G is the downward heat flux in the soil surface; u is the wind speed; T is the air temperature; RH is the relative humidity; and n are the sunshine hours.

**Figure 3.**Calculation flowchart of the crop module. The EAI is the ear area index; FT, FC, FN, and FH are the temperature influence factor, CO

_{2}concentration influence factor, nitrogen influence factor, and water influence factor, respectively.

**Figure 5.**Schematic diagram of energy distribution in soil surface and the canopy resistance of the hydrothermal transmission.

**Figure 6.**Field experiment plot layout. The open circle (○) denote the position of the neutron probe; the solid circle (●) denote the position of soil temperature thermistor; the plus symbol (+) denote the crop growth index monitoring.

**Figure 7.**Comparison between simulated and measured soil water storage in 0–1 m soil layer in calibration treatments. The error bars are the standard deviation of field measurements in the two or four medium fertilization plots, as shown in Figure 6 (e.g., the error bars in W0 is calculated from W0FM-1 and W0FM-2; W2 from W2-1FM-1, W2-1FM-2, W2-2FM-1, and W2-2FM-2). The P and I represent precipitation and irrigation.

**Figure 8.**Comparison between simulated and measured soil water storage in 0–1 m soil layer in validation treatments.

**Figure 9.**Comparison between simulated and measured soil water content of different soil layer in W4 treatment. (

**a**–

**d**) is March 25, April 27, May 20 and June 2, respectively.

**Figure 10.**Comparison between simulated and measured soil water content of different soil layer in W1 treatment. (

**a**–

**d**) is March 25, April 27, May 20 and June 5, respectively.

**Figure 11.**Comparison between simulated and measured soil temperature at soil depth 10 cm, 40 cm, 60 cm under different water treatments.

**Figure 18.**Comparison between CropSPAC and SPAC model simulated soil water storage in 0–1 m soil layer in W0 treatment.

Treatment | Fertilization (kg/ha) | Irrigation Volume (mm) | Total Irrigation (mm) | |||
---|---|---|---|---|---|---|

21 April 1999 | March 26 | April 21 | May 4 | May 19 | ||

W0 | 150 | - | - | - | - | 0 |

W1 | 150 | - | 60 | - | - | 60 |

W2 | 150 | - | 60 | - | 50 | 110 |

W3 | 150 | 60 | 60 | - | 50 | 170 |

W4 | 150 | 60 | 60 | 60 | 50 | 230 |

Initial Conditions | Symbol | Unit | Value |
---|---|---|---|

Physiological development time | PDT | 8 | |

Vernalization | VP | 1 | |

Dry biomass amount | W_{day} | kg/ha | 2585 |

Dry aboveground biomass amount | W_{top} | kg/ha | 2226 |

Dry leaf biomass amount | W_{leaf} | kg/ha | 601 |

Soil Texture | K_{s} (cm/min) | θ_{s} | θ_{r} | α (cm^{−1}) | n |
---|---|---|---|---|---|

Sandy loam | 0.02 | 0.48 | 0.05 | 0.02 | 1.34 |

_{s}is the saturated water content; θ

_{r}is the residual moisture content; and α and n are the parameters in the VGM model.

Treatments | Measured Yield (kg/ha) | Simulated Yield (kg/ha) | Relative Error |
---|---|---|---|

W0 | 3395 | 3987 | 17.4% |

W1 | 4722 | 4192 | 11.2% |

W2 | 5378 | 4518 | 15.9% |

W3 | 5641 | 4925 | 12.6% |

W4 | 5340 | 4937 | 7.5% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, W.; Mao, X.; Yang, J.; Ji, M.; Adeloye, A.J.
A Coupled Model for Simulating Water and Heat Transfer in Soil-Plant-Atmosphere Continuum with Crop Growth. *Water* **2019**, *11*, 47.
https://doi.org/10.3390/w11010047

**AMA Style**

Yang W, Mao X, Yang J, Ji M, Adeloye AJ.
A Coupled Model for Simulating Water and Heat Transfer in Soil-Plant-Atmosphere Continuum with Crop Growth. *Water*. 2019; 11(1):47.
https://doi.org/10.3390/w11010047

**Chicago/Turabian Style**

Yang, Weicai, Xiaomin Mao, Jian Yang, Mengmeng Ji, and Adebayo J. Adeloye.
2019. "A Coupled Model for Simulating Water and Heat Transfer in Soil-Plant-Atmosphere Continuum with Crop Growth" *Water* 11, no. 1: 47.
https://doi.org/10.3390/w11010047