Experimental data from an aeroponic system were selected to evaluate the described approach to model the influence of temperature on the vertical penetration of the root system. The experimental setup in an aeroponic system with no soil and no water movement has the advantage of evaluating the effects of temperature on root growth while limiting the effects of other factors. The experiment was conducted in a greenhouse with bell pepper at the Jacob Blaustein Institutes for Desert Research, in Midreshet Ben-Gurion, Israel . The objective was to evaluate the effect of three different root zone temperatures on root and plant growth. Six bell pepper plants were cultivated from 7 Jan. to 20 Feb. 2015 in aeroponic pots mounted on top of the aeroponic systems. The aeroponic apparatus comprised circular pots made from plastic material with a diameter of 50 cm and a depth of 14 cm. Within each thermally isolated aeroponic pot, misters were fixed to produce the desired fine mist sprayed directly onto the plant roots. The computer-controlled spraying varied from 8-s sprayings at 1-min intervals, depending on the growth stage of the plants and the temperature of the greenhouse. Three different water temperatures were applied to each treatment, namely 7, 17, and 27°C. The air and root zone temperatures were measured daily. The air temperatures in the greenhouse were 25°C during the day and 18°C at night. The treatments were replicated twice, leading to six tanks in total. The maximum rooting depth of each plant was observed four times during the 44 d of plant growth. In the aeroponic systems,rolling bench all other factors that affect the growth and development of roots were rendered insignificant during the treatment.

The implemented modeling approach to simulate the temperature dependent vertical root penetration was evaluated by comparing modeling results with the measured maximum rooting depths in the experimental aeroponic system with bell pepper. Because two approaches were implemented in HYDRUS to describe both time-dependent potential root growth and the temperature stress factor, four combinations are thus available to describe the temperature-dependent root growth. All four models were tested against measured maximum rooting depths to validate their ability to describe the temperature- and time-dependent vertical root penetration. The four combinations are summarized in Table 3. Were the these root growth models able to properly simulate the influence of temperature on vertical root penetration, a single combination of model parameters for each model that could reproduce the measured maximum rooting depths for all temperature treatments would have to exist. An overview of the model parameters that have to be specified for each model is given in Table 4. Table 4 shows that, depending on the model, four or six parameters have to be specified to model temperature-dependent root growth. In a complex soil water flow model such as HYDRUS, four to six additional parameters can significantly increase the calibration effort and parameter uncertainties. The temperature-dependent modeling approach was tested outside of the HYDRUS implementation and within a MATLAB environment. The goal of the evaluation was to determine whether the combination of the time-dependent root growth functions and the temperature stress functions were able to reproduce the measured rooting depths under the given boundary conditions.

A global sensitivity analysis was conducted using the Sobol¢ method to reveal the key parameters of each model and to determine the contribution of the uncertainty of each parameter to the uncertainty of the model output. The Sobol¢ method is based on variance decomposition and provides the impact of each parameter and its interactions with other parameters on the model output . This type of global sensitivity analysis can be applied to nonlinear and non-monotonic models and is a widely used tool for sensitivity analysis studies. Its ability to account for interactions between model parameters is an important advantage of the Sobol¢ method .An overview of studies using the Sobol¢ method for sensitivity analysis in hydrological modeling was provided by Song et al. . The method has already been applied with the HYDRUS software package by Li et al. , Brunetti et al. , and Wang et al. . Sobol¢ proposed that the total variance of the model output can be decomposed into component variances of individual parameters and their interactions. The first-order sensitivity index quantifies the main effect of the ith parameter, Xi . This sensitivity index denotes the part of the total variance due to Xi without considering the interactions with other parameters. The total-order sensitivity index additionally includes the proportion of the variance due to the interactions of Xi with the other parameters. The values of the indices vary from 0 to 1, where 0 stands for no influence and 1 for a high influence on the variance.The number of parameter sets in the sensitivity analysis of the four models was set to 10,000. This number was initially set higher than in studies of Brunetti et al. and Zhang et al. to avoid a time-consuming convergence analysis of the sensitivity analysis and to achieve a higher accuracy of the sensitivity analysis, which increases with an increasing number of model runs.

The p parameter depends on the considered model and is either four or six . To calculate the sensitivity indices for all i = 1, …, p parameters, Matrix AB i has to be evaluated p times. The total number of model runs required to calculate the sensitivity indices for all parameters of each model were M = N. Archer et al. suggested using bootstrap confidence intervals to evaluate a suitable accuracy of the sensitivity estimates. For this reason, each estimation of the sensitivity indices was repeated 500 times to evaluate the 25th and 75th percentiles of the sensitivity indices. The small number of repetitions is due the fact that the sensitivity analysis of four models, each including four or six parameters, requires a high level of computational effort. Evaluated percentiles were used only as an additional parameter to assess the sensitivity indices .Rather than using experimental data, which may be subject to various errors and effects of various factors, the sensitivity analysis was performed using a hypothetical data set in which observed rooting depths LObs were generated by running all four models with a predefined parameterization. The cardinal temperatures Tmin, Topt, and Tc were set to 8, 23, and 27°C, respectively. A complete list of parameter values for the model runs is provided in Table 5. The boundary conditions were similar to those of the aeroponic experiments. The temperature was set to 22°C during the first 14 d and to either 7, 17, or 27°C during the remaining time period of 150 d. Additionally,grow table hydroponic a fourth scenario was considered with a temperature of 37°C after the first 2 wk to make sure that the specified cardinal temperatures would lie within the applied boundary conditions to determine their influence on the model output. Each model was thus executed four times with four different temperature boundary conditions to generate data for Lsim in Eq..The Differential Evolution Adaptive Metropolis algorithm was used for optimizing the model parameters and for model calibration. The DREAM algorithm is based on Bayesian statistics; it runs multiple different Markov chains to generate a random walk through the search space. Based on a proposal distribution, the sampler evolves to the posterior distribution by iteratively finding solutions with stable frequencies stemming from the fixed probability distribution . The Gaussian likelihood function was used to summarize the distance between model simulations and corresponding observations. The residuals were assumed to be independent and normally distributed while the measurement error was neglected.Eight Markov chains were run with a set of 5000 generations. The initial state of each chain was sampled from a Latin hypercube. The parameter space of each parameter was defined by using the same boundaries as were used for the sensitivity analysis . These parameter limits also define the search domain for the predominantly physically based parameters . The calculation time for the DREAM optimization of a single model was approximately 10 min, with no parallelization needed on a machine with the following specifications: Intel Core i7–4710HQ CPU with 2.50 GHz of RAM and 12 GB of storage.We evaluating the effects of different factors on root growth with two examples using the new root growth module in HYDRUS-1D and HYDRUS-2D.

Then we analyzed the collected experimental data and the evaluation of the temperature-dependent root growth modeling approach. Therefore, we first collected data on the experimental outcomes. Second, we carried out the sensitivity analysis to evaluate the sensitivity of the modeling results to various input parameters and identified which parameters need to be fitted and which can be set to values from the literature. Third, we used the DREAM optimization approach to analyze the collected experimental data while considering the results of the sensitivity analysis.We used two hypothetical examples that demonstrate the implemented root growth model and the impact of various environmental factors on root growth. In the first example, we used HYDRUS-1D and simulated optimal root growth as well as root growth restricted due to low water availability, temperature, texture, and bulk density. In the second example, we used HYDRUS-2D to again simulate optimal root growth and then root growth affected by a nonuniform distribution of water contents due to asymmetrical irrigation.Figure 2 shows examples of the development of simulated root systems under the influence of various environmental factors compared with the potential development of the root system, independent of environmental factors . The soil profile was considered to be homogenous, consisting of 10% sand, 50% silt, and 40% clay and having the bulk density of 1530 kg m−3. The parameters for the soil hydraulic functions of van Genuchten and Mualem were estimated from the textural information using the Rosetta module of HYDRUS-1D. Time-dependent root growth was simulated using the function of Borg and Grimes . Model 2 from Table 2 was selected to describe the potential root length density distribution. The maximum potential rooting depth was set to 120 cm, and root growth was considered for 90 d . The upper and lower boundary conditions were set to atmospheric boundary conditions with surface runoff and free drainage, respectively. In Scenario 1, the development of the root length densities was simulated for drought conditions, which may cause aeration stress and root senescence . Potential evaporation and transpiration rates were set to increase stepwise at the beginning and remain constant at the end of the simulation. Irrigation took place at irregular intervals and covered only 25% of the potential evapotranspiration. Due to low water availability, the root system was underdeveloped compared with the reference simulation when no restrictions on root growth were considered . The root length density in this scenario is reduced compared with Scenario 0, mainly close to the soil surface because the drought stress is highest in soil horizons with the highest root water uptake. Scenario 2 considered the effects of texture and bulk density on the development of the root length densities and vertical root penetration. The results show that due to the increased soil strength , the potential maximum rooting depth of 120 cm was not reached. The additional stress due to the soil strength negatively influenced the vertical penetration of roots, in addition to the effects of root senescence due to the drought stress in Scenario 1. Scenario 3 considered the influence of different root zone temperatures on the root length density development. In this case, only the option to consider the influence of temperature on root growth was enabled. The development of the root system was simulated using the root growth function of Borg and Grimes in combination with the temperature stress function of Jones et al. . The parameters Tmin, Topt, and t m were set to 5°C, 35°C, and 90 d, respectively. The temperature throughout a soil depth of 150 cm was set to a constant value of 10°C. The temperature at the soil surface was set to 25°C during the first 14 d and to 30 and 7°C in Scenarios 3a and 3b, respectively, until the end of the simulation. The results show that the root system in Scenario 3b, which was exposed to temperatures close to Tmin, was underdeveloped compared with the root system in Scenario 3a, which was exposed to temperatures close to Topt.