The root system under conditions that were close to the optimum temperature showed greater growth rates and thus reached deeper depths and developed greater root length densities than the root system under non-optimum temperature conditions. Finally, an additional soil horizon, consisting of 24% sand, 64% silt, and 12% clay and having the bulk density of 1290 kg m−3, was specified between 20 and 50 cm in Scenario 4. The same stress factors as in Scenario 2 were enabled. The results for this scenario show that, due to the different properties of the additional soil horizon, which affect water availability and O2 supply, the development of the root length density distribution can be highly influenced. The results show that the root length density can be spatially altered due to a spatial distribution of stress factors. In this case, unfavorable growing conditions led to a reduction of root length density in the middle layer.Figure 3 shows the simulated potential and actual two-dimensional development of the root system in a homogeneous soil consisting of 50% sand, 20% silt, and 30% clay and having a bulk density of 1510 kg m−3. The parameters for the soil hydraulic functions of van Genuchten and Mualem were estimated using the Rosetta module of HYDRUS-2D. The lower boundary condition was set to free drainage. The simulation time was set to 60 d, and root growth was considered until the end of the simulation. Potential transpiration was set to 0.1 mm d−1 between the first day and Day 11. From Days 12 to 29, potential transpiration was set to 1 mm d−1; from Day 30 until the end of the simulation, it was set to 2 mm d−1. Irrigation was applied at a rate of 2 mm d−1 between Days 30 and 50 only in the top left corner ,flood and drain table while the rest of the soil surface remained unirrigated. Potential evaporation was specified only for the unirrigated soil surface, with a rate of 0.1 mm d−1 from the beginning until Day 15, 1 mm d−1 between Days 16 and 18, and 2 mm d−1 from Day 19 until the end of the simulation.

In addition to water availability and aeration, the texture and bulk density were also considered influences on root growth. Only one plant in the middle of the domain was specified with a potential rooting depth in the vertical direction of 100 cm and 30 cm in the horizontal direction. The root length density function of Vrugt et al. predicts an axially symmetrical shape of the potential root length density distribution.The root system is more developed on the left hand side of the transport domain than the right-hand side due to greater water availability. Additionally, the root system simulated under the influence of growth restrictions showed a smaller rooting depth than the reference simulation when no restrictions on root growth were considered.The results of the global sensitivity analysis in the form of the totaland first-order sensitivity indices of all parameters of the four root growth models are shown in Fig. 4. The light gray bars represent the contribution of each individual parameter to the variance of the model output . The dark gray bars represent the individual parameter contributions, as well as the contributions of their interactions with the other parameters to the variance of the model output. The results of the sensitivity analysis reveal that the differences among the influences of different parameters in Models A and C are more pronounced than between different parameters in Models B and D. The differences between the sensitivity indices of different parameters of Models B and D are not very different from each other.

Figure 4 reveals that the plant-specific cardinal temperatures Topt, Tmin, and Tc and the plant-specific parameter Lm have only a minimal impact on the model outcome. Because these parameters are biologically based and specific for a particular crop cultivar, their values can be found in the literature and do not have to be calibrated. For example, Yan and Hunt provided cardinal temperatures for sorghum [Sorghum bicolor Moench], wheat, barley, bean , and maize, and Li et al. provided a list of sources for cardinal temperatures for winter wheat. Borg and Grimes provided an overview of Lm for several crops. Specifying parameters that do not exert a large influence on the model output using values found in the literature reduces the number of calibrated parameters and thus the complexity and effort of the calibration task. By defining a limit of 0.3 for the total sensitivity index to neglect the parameter influence, the number of calibrated parameters of Model A can be reduced from four to two. Based on the results in Fig. 4, the number of calibrated parameters of Model B can be reduced from six to one. However, because no information about parameters r and ct could be found in the literature, these parameters need to be considered within the model calibration. The number of calibrated parameters could thus only be reduced to two for Model B. Yan and Hunt suggested that reasonable results can be found by setting ct to 1. They also suggested that it may be possible to set Tmin to 0°C, which could be acceptable for all crops except for summer crops. The results of the global sensitivity analysis for Models C and D show that the parameter Lm is less important than in Models A and B. In these models, the parameter defining the time when plants reach maturity, t m, demonstrates a high influence on the model output. Based on the results of the sensitivity analysis, the number of calibrated parameters can be reduced from four to a single parameter for Model C and from six to three for Model D, while the other parameters can be specified using values from the literature. In contrast to the other models, the results presented in Fig. 4 suggest the parameter Tc should be considered as a calibration parameter for Model D.

Due to the lack of knowledge about the value of ct, this parameter should still be considered in the model calibration.The optimized parameter values of all four modeling approaches are listed in Table 6. In addition to these,hydroponic equipment the table also provides an overview of the goodness of fit obtained by individual models determined by the RMSE. This value indicates how well the rooting depths simulated by the models using the optimized parameter sets fit the observed maximum rooting depths. Figure 5 visualizes the maximum rooting depths measured for all temperature treatments in the aeroponic system and simulated using the four models and the optimized parameter sets in Table 6. The upper left figure shows the simulation results for all four models compared with the average observed maximum rooting depths in each tank. These averaged values were used as data in the comparison within the optimization task. The other three subplots show a comparison between the simulated rooting depths and the measured maximum rooting depths of all six plants for all three temperature treatments. Figure 5 and the RMSE in Table 6 indicate that Models C and D fitted the observed data very well. Models A and B did not seem to capture the root growth dynamics as well as the other two models. This is underscored by the RMSE values listed in Table 6. The graphs produced by Models A and B are quite similar, and both resulted in an RMSE of 7 to 8 cm. In contrast, Models C and D produced RMSE values below 4 cm. The best correspondence between the model and the data, with the smallest RMSE of only 2 cm, was achieved with Model D, which uses the growth function of Borg and Grimes and the stress reduction function of Yin et al. . However, despite the slightly worse performance of Model C, the results show that both temperature stress approaches can be used to simulate temperature-dependent root growth when used in combination with the time-dependent root growth function of Borg and Grimes . Table 7 lists the optimization results when the numbers of calibration parameters were reduced based on the results of the sensitivity analysis. In this case, the parameters with a total-order sensitivity index lower than 0.3 were set equal to values from the literature. The parameters describing cardinal temperatures were taken from Reddy and Kakani , who investigated cardinal temperatures of bell pepper for pollen germination. Due to the low sensitivity of cardinal temperatures on the model output, these temperatures could also be used to model the root growth, even though this has not yet been scientifically documented. The value for Lm was taken from Borg and Grimes . Based on the results of the sensitivity analysis, the parameter Tmin was still considered as a calibration parameter in Models A and B.

The optimization results shown in Table 7 indicate that despite the reduction in the number of calibrated parameters, the evaluated root growth models, especially Models C and D, produced an acceptable correspondence with the measured data and associated goodness of fit. The other two models show a larger increase in the RMSE, which indicates a drop in the goodness of fit. Simulated rooting depths for all temperature treatments using the four root growth models and the optimized parameter sets from Table 7 are shown in Fig. 6. Figure 6 and the RMSE in Table 7 indicate that Model D also fit the observed data very well, even when three of the six model parameters were specified using values taken from the literature and not calibrated. Model C, which only had one calibration parameter, provided a sufficient reproduction of the observed maximum rooting depths.We implemented an adapted root growth modeling approach based on Jones et al. in the HYDRUS software packages. The modeling approach describes root growth and root system development under the influence of environmental factors such as soil hydraulic properties, soil strength, water availability, chemical conditions, and temperature. The implementation of growth and stress functions in the HYDRUS software opens the opportunity to derive parameters of these functions from laboratory or field experimental data using inverse modeling. This option was then demonstrated using experimental data from an aeroponic system for the temperature stress part of the root growth model. The impact of temperature on root growth was described using two stress response functions, both combined with two, time-dependent root growth functions. The four resulting combinations were tested against rooting depth data measured using an aeroponic system experiment. Because the experiment was not initially set up to evaluate the described root growth module, the collected data were not fully suitable to identify model parameters with high accuracy. For example, the use of a wider temperature range would help to better identify cardinal temperatures and the maximum potential rooting depth . Nevertheless, the results indicate that the temperature-dependent root growth approaches are well capable of reproducing real root growth data. All four of the tested root growth models reproduced the measured rooting depth data very well, with RMSE values ranging between 2.1 and 8 cm. Especially the results of the root growth model referred to as Model D, which combined the Borg and Grimes root growth function with the Yin et al. temperature stress function, showed an excellent agreement with the observed rooting depth data. The results indicate that the implementation of the time-dependent root growth function of Borg and Grimes into the HYDRUS software was beneficial because this function captured the actual root growth better than the originally implemented Verhulst–Pearl logistic growth function. However, a complete evaluation of the two proposed approaches simulating root growth still requires further testing under dynamic stress conditions that would occur under field conditions, e.g., by using data provided by Aidoo et al. . A sensitivity analysis revealed that the biologically based parameters of the models, such as the cardinal temperatures and the maximum rooting depth, tend to have a small impact on the model outcome. Because of the low sensitivity of the model outputs to the uncertainties originating from these parameters, it seems reasonable to approximate these parameters with values from the literature and to reduce the number of parameters that need to be calibrated.