Fundamentally, this ecohydrological model is an integration of different components from the Soil and Water Assessment Tool (SWAT) (Neitsch et al., 2011; Arnold et al., 2012) and the Community Land Model version 4.5 (CLM4.5) (Oleson et al., 2013) (Fig. 1). On this basis, root growth and root uptake modules were modified. Furthermore, an optimization approach was introduced to simulate the coupled effects of root growth and soil water dynamics.

Hydrologically, surface process modules, for example, simulating evaporation, transpiration, canopy interception, and runoff, follow the SWAT model, which is detailed in its theoretical document (Neitsch et al., 2011). The subsurface hydrological modules, primarily the soil water movement, adopted the 1-D Richards' equation. It is solved numerically following the finite difference scheme used in CLM4.5 (Oleson et al., 2013).

Biologically, modules for the above-ground parts, for example, plant phenology, leaf area index (LAI) development, and biomass accumulation, are adopted from SWAT (Neitsch et al., 2011) and CLM4.5 (Oleson et al., 2013). The module for the below-ground part, the root growth model, simulates root growth following CLM4.5 which, in turn, adopts the approaches from process simulation directly from the implementation in the Biome-BGC (Biome BioGeochemical Cycles) model (Thornton et al., 2002). See the Supplement for more information on the detailed model descriptions. In CLM4.5, the roots are categorized into coarse and fine. The rooting depth is presented as a static value, and the fine-root distribution follows a static shape function which defines the root density over the soil profile.

The root–water interaction is integrated into the root uptake model, which uses the rooting depth and fine-root distribution as input and acts as a sink term in the Richards' equation. Soil water imposes water stress on the root growth. The cost of biomass invested in coarse and fine roots and the benefit of water uptake were optimized through a cost–benefit function.

The following sections will describe three root growth approaches in a stepwise manner: (1) the static root distribution approach implemented in CLM4.5, which assumes a static rooting depth of the coarse roots and a static distribution of fine roots (Oleson et al., 2013); (2) the approach that assumes a dynamic distribution of fine roots but a static rooting depth (Drewniak, 2019); and (3) the approach proposed in this study that assumes a dynamic distribution of fine roots and a dynamic rooting depth of coarse roots. These three root growth modelling approaches are incorporated into the ecohydrological model mentioned above for a comparison of their performances.

In this approach, a static exponential function expresses the root fractions in different soil layers (Zeng, 2001): 1fri=0.5exp⁡-razh,i-1+exp⁡-rbzh,i-1-exp⁡-razh,i-exp⁡-rbzh,i1≤i<n0.5exp⁡-razh,i-1+exp⁡-rbzh,i-1i=n, where i is the sequential number of soil layers, n is the total number of soil layers in the rooting zone, zh,i is the depth of soil layer i, and ra and rb are two shape parameters. The shape parameters can be obtained by fitting to the observations for different plant types and are set as 4.8 and 0.8 according to fine-root sampling in this study (see Sect. 2.2.4) respectively (Oleson et al., 2013).

This approach assumes that newly assimilated biomass to the below-ground parts is allocated to develop fine and coarse roots with a static ratio. In general, there is a linear relationship between the carbon mass and biomass (Niu et al., 2020). Therefore, the term “carbon” will be used in the following text to refer to the biomass of different plant components. Within the present rooting depth (which is static over the simulation period), the fine roots are distributed over the soil depth according to the soil water content. In each soil layer i, the fine-root carbon increment is updated at each time step, as follows: 2FRi,t=FRi,t-1+ΔFRi,t, where FRi,t-1 (g m-2 d-1) is the fine-root carbon of soil layer i during the previous time step, and ΔFRi,t (g m-2 d-1) is the newly allocated carbon to fine roots of soil layer i at time t. ΔFRi,t is modified by the soil water content as follows: 3ΔFRi,t=ΔFR⋅THKiREWi∑i=1nTHKiREWi, where ΔFR is the newly assimilated carbon allocated to total fine roots (g m-2 d-1), n is the total number of soil layers in the rooting zone, THKi is the soil layer thickness of soil layer i (cm), and REWi is the relatively effective soil water content (i.e. soil water availability). REWi is calculated as follows: 4REWi=θ-θwpθfc-θwp, where θ is the soil water content (cm3 cm-3), and the subscripts “wp” and “fc” indicate the soil water content at the wilting point and field capacity respectively.

The fine-root fraction in each soil layer i is then calculated: 5fri=FRi,t∑i=1nFRi,t.

This approach assumes that both the rooting depth of coarse roots and the distribution of fine roots change with soil water. In formulating the growth of the coarse and fine roots, the newly allocated biomass/carbon for the below-ground part is optimally allocated to the roots based on a cost–benefit function that will be described in detail.

The cost is defined as the amount of carbon invested to grow coarse/fine roots. In constructing the coarse-root and fine-root systems, the pipe model theory (PMT; Shinozaki et al., 1964) was adopted. Lehnebach et al. (2018) summarized “the essence of the PMT concept” as “a unit amount of leaves is provided with a pipe whose thickness or cross-sectional area is constant. The pipe serves both as the vascular passage and as the mechanical support and runs from the leaves to the stem through all intervening strata.” The relationship between the leaf and stem can be established quantitatively based on the PMT and can be extended to the below-ground parts of the plants (Chen et al., 2019). For roots, the relationships have been established in analogue form and validated against some databases (Carlson and Harrington, 1987; Richardson and Dohna, 2003).

Delineating the soil profile into adjacent soil layers which correspond to the numerical solution of the Richards' equation, the equations for modelling the root growth are also written regarding the discrete soil layers (Fig. 2c).

The carbon amount for coarse and fine roots should be maintained as 6CR0i=ρ⋅Zi⋅kA⋅FRi, where CR0i is the equivalent carbon for coarse roots in the zone between the ground surface and bottom of soil layer i, FRi is the carbon for fine roots within soil layer i, ρ is the mass density of coarse roots (g cm-3), Zi is the depth from the surface to the bottom of soil layer i (cm), and kA is a constant which can be determined from field observations (see Sect. 2.3.2).

For an increment of carbon for fine roots within soil layer i, a corresponding increment of carbon for coarse roots is needed and can be derived from Eq. (6): 7ΔCR0i=ρ⋅Zi⋅kA⋅ΔFRi. Notably, there is no increment of carbon for coarse roots if the available coarse roots are sufficient to support the new fine roots, as discussed later.

The increments of carbon for fine and coarse roots were then summed over the root profile: 8ΔCR=∑1nΔCR0i,9ΔFR=∑1nΔFRi. The sum of the newly allocated carbon for fine and coarse roots is equal to that allocated to the below-ground part ΔTR: 10ΔFR+ΔCR=ΔTR. Two respective ratios are defined for fine and coarse roots as follows: 11KFR=ΔFRΔCR+ΔFR,KCR=ΔCRΔCR+ΔFR.

Potentially, more fine roots develop in wetter soil zones in order to gain more water and to reduce water stress as much as possible. It is fundamentally recognized that penetration into deeper soil requires carbon for the fine roots, as well as the corresponding coarse roots; the potential benefit can be more water uptake from the deeper soil. The basic principle for the distribution of the roots, either fine only or both coarse and fine, is that an optimal distribution of new roots helps to gain as much water as possible (Fig. 2d).

Thus, the distribution of fine roots is influenced by a cost–benefit ratio, which is defined as follows: 12Wi=REWiCFRi, where Wi is the cost–benefit ratio in soil layer i, and the benefit is presented by REWi, as defined previously. CFRi is defined as the marginal carbon cost of fine roots: 13CFRi=∂ΔCR0i+ΔFRi∂ΔFRi=ρ⋅Zi⋅kA+1. Combining Eqs. (13) and (12), 14Wi=REWiρ⋅Zi⋅kA+1. Replacing REWi with Wi in Eq. (3), the fraction of new fine roots in the soil layer i becomes 15ΔFRi=ΔFR⋅THKiWi∑i=1mTHKiWi. Updating the fine-root fractions using Eq. (7), the demand for coarse roots that can meet the hydraulic transport demand for the fine roots within and below soil layer i is 16CRPi=∑j=inCR0i,NEW=∑j=inρ⋅Zi⋅kA⋅FRi+ΔFRi. Comparing the demand for and current storage of coarse roots, the coefficient of the coarse-root carbon increment regarding the soil layer i is calculated as 17βi=maxCRPi-CRi,0∑i=0nmaxCRPi-CRi,0. According to Eq. (11), the increment of carbon for the coarse roots with respect to each soil layer is then calculated as 18ΔCRi=ΔTR⋅KCR⋅βCi.

To calculate the rooting depth extension via optimization, the target function Sm is defined as follows: 19Sm=∑i=1mTHKi⋅fri⋅REWi. When Sm reaches its maximum, the optimum distribution of new roots over the soil profile is obtained. If Sm reaches its maximum when m is equal to n, the length of the coarse root remains unchanged; however, when m is equal to n+1, the coarse root penetrates the next soil layer.

The study site is located in the Yeheshan Provincial Nature Forest Reserve (34∘31.76′ N, 107∘54.67′ E; 1090 m elevation) in the southern part of the Loess Plateau in China (Fig. 3). The climate is semi-humid with an annual average temperature of 11.3 ∘C and precipitation of 570 mm (from Yongshou station; see Sect. 2.2.2). It is hot in summer and cold in winter, and precipitation occurs predominantly from May to October with significant inter-annual variation. Artificially afforested black locust (Robinia pseudoacacia L.) dominates the vegetation species with an average height of 10 m (planted in 2000) and a density of 2450 trees ha-2 (Ma et al., 2017). The experimental plots were situated within a black locust forestland on an average slope of 8∘. Instruments for the microclimate and soil water observations were installed. The thickness of the loess soil is estimated to be more than 50 m, and the buried depth of groundwater is beyond that depth (Liu et al., 2010).

A meteorological observation system was established on a flux tower in 2014. The tower was 16 m above the ground, higher than the tree canopy (Fig. 3). This system consists of HMP155A probes (Campbell Scientific, Inc., Logan, UT, USA) for temperature and humidity, a CSAT3 3-D ultrasonic anemometer (Campbell Scientific, Inc., Logan, UT, USA) for wind speed, and a CNR4 net radiometer (Campbell Scientific, Inc., Logan, UT, USA) for solar radiation. A T-200B precipitation gauge (Geonor Inc., Oslo, Norway) was installed near the forest opening to measure throughfall. A CR3000 data logger (Campbell Scientific, Inc., Logan, UT, USA) was used to collect data from the sensors at 10 min intervals.

In addition, daily meteorological data for 1971 to 2020 from the National Metrological Station in Yongshou County, 26 km from the field experiment site, were downloaded from the China Meteorological Data Service Centre (http://data.cma.cn, last access: 23 March 2021) for the long-term simulation. The data series include daily precipitation, mean air temperature, maximum air temperature, minimum air temperature, relative humidity, wind speed, and sunshine hours.

It is believed that the data series, spanning 50 years, covers the inter-annual variations in climatic factors, especially the alternating wet and dry periods (Fig. 4). The annual average precipitation is 570 mm, with a standard deviation (SD) of 122 mm.

The soil properties were investigated by sampling over a profile 5.0 m below the ground at the study site. The dry bulk density (ρb, g cm-3) was obtained by drying volumetric soil samples (100 cm3) at 105 ∘C for 48 h, and the soil particle size distribution and organic matter content were measured in the laboratory. Silt loam dominated the profile, with moderate variations among the soil layers. On average, the silt loam consisted of 5.8 % sand, 73.4 % silt, and 20.9 % clay (Ma et al., 2017).

The saturated hydraulic conductivity (Ks, cm h-1) was measured in the undisturbed soil samples using a constant-head method (Ramos et al., 2017). The field capacity (θfc) and wilting point (θwp) were derived from the power function (Campbell, 1974; Clapp and Hornberger, 1978), corresponding to the soil potentials of -33 and -1500 kPa respectively.

Volumetric soil moisture sensors were installed in 14 layers within a depth of 500 cm from the surface: at 5, 15, 30, 45, 60, 90, 120, 150, 200, 250, 300, 350, 400, and 500 cm respectively (Fig. 3). The sensors, CS-655 soil water content reflectometers (Campbell Scientific, Inc., Logan, UT, USA), have been in operation since June 2014. A CR1000 data logger (Campbell Scientific, Inc., Logan, UT, USA) records the data every 10 min.

The soil properties below 5.0 m were adopted from previous studies (Li et al., 2008; Jia et al., 2017a; Wu et al., 2021) on black locust plantations in the Loess Plateau.

The soil desiccation index (SDI) was used to evaluate the degree of soil desiccation for the comparison between the observation and simulation results; this index was calculated as follows: 20SDI=θ-θwpθsfc-θwp, where θsfc is the soil water content at a stable field capacity. In practice, a soil water content at 60 % of field capacity (θfc) can be assumed to be the stable field capacity of loess in the Loess Plateau (Wang et al., 2011). Soil layers with an SDI < 1 were regarded as drying soil layers.

The leaf area index (LAI, m2 m-2) of the black locust canopy was measured using an optical method (Jonckheere et al., 2004), biweekly or triweekly during the growing seasons of 2014, 2015, and 2016. An 8 mm fisheye lens (Sigma F3.5 EX DG circular fisheye, Sigma Corporation) mounted on a Canon EOS 5D digital SLR camera (http://www.canon.com, last access: 20 December 2020) took hemispherical photographs of the canopy on cloudy days. The photographs were analysed using the CAN_EYE software to derive the LAI values (Demarez et al., 2008). Within each plot, photographs were taken at five different positions each time. The LAI value for the plot is the average of the five positions.

The measured LAI values were compared to those of the Moderate Resolution Imaging Spectroradiometer (MODIS) MYD15A2H product, which has a spatial resolution of 500 m and a temporal resolution of 8 d. Furthermore, the 8 d MODIS LAI series were downscaled to a daily series using the Savitzky–Golay filtering technique (Tie et al., 2017).

The root profiles were investigated in August 2015. A 150 cm wide trench, which was located between two neighbouring rows and perpendicular to the row direction, was excavated 500 cm below the ground (Fig. 3). During the excavation, soil samples, 20 cm in the horizontal dimension and 40 cm in vertical thickness, were taken along the trench and over the profile. It is assumed that the roots develop homogeneously along the horizontal row but unevenly along the trench. Finally, seven samples were collected from each soil layer across the trench. The soil samples were rinsed, and the weight and length of fine roots (less than 2 mm in diameter) were measured for each sample.

To estimate the parameters of the black locust root system, a pot experiment was carried out in 2016 (planted in April and sampled in October), the details of which are given in Fig. S1 in the Supplement.

In addition to the measurements mentioned above, data on black locust roots, including the density profile and rooting depth were compiled from the published literature (see Supplement).

In solving the Richards' equation numerically, the vertical domain extends from the surface to 20 m below the ground. The domain was discretized into adjoining layers with a thickness of 5 cm each.

The upper boundary condition was set as the flux of the rainfall rate with the canopy interception removed or the soil evaporation rate. As the soil water content in deep soil (20–100 m) is relatively stable around the field capacity (Qiao et al., 2018), the lower boundary was set to a constant soil water content at the field capacity.

During the calibration and validation stages, the initial soil water profile was determined using the measurements. The initial soil water profile was set at the field capacity when the model was applied to the long-term simulation.

The soil hydraulic parameters, saturated hydraulic conductivity Ks and constant B in the soil water retention curve, were initialized by the measured values. They were further tuned to match the simulated soil water content to the observations.

The vegetation growth parameters were adapted from Zhang et al. (2015), who simulated the black locust growth in the Loess Plateau using the Biome-BGC model. Other vegetation parameters were obtained from the SWAT model (Neitsch et al., 2011; Sun et al., 2011).

All the relevant parameters are summarized in Table S1 in the Supplement.

The numerical simulations consist of two parts, the short-term (5 years) simulation for model calibration and validation, and the long-term simulation (50 years) for investigation:

The model calibration and validation were performed for the observation period (2014–2018). The model was calibrated from 1 June 2014 to 31 December 2016 and validated through 1 January 2017 to 31 December 2018. The measured LAI, soil water content, and root profiles were used for the evaluation. The rooting depth was assumed to be 5.0 m below the ground for three approaches, considering the relatively short period of the field experiment.

Statistical indices, the coefficient of determination (R2), the Nash–Sutcliffe efficiency (NSE), and the percent bias (PBIAS), were used to evaluate model performance and are given as follows: 21R2=∑i=1nSi-S‾Oi-O‾∑i=1nSi-S‾∑i=1nOi-O‾;22NSE=1-∑i=1nOi-Si2∑i=1nOi-O‾2; 23PBIAS=∑i=1nSi-Oi∑i=1nOi×100%. Here, Si is the simulated value at time step i, S‾ is the mean of the simulated value, Oi is the observed value at time step i, O‾ is the mean of the observed value, and n is the number of time steps. R2 and NSE were dimensionless. The dimension of the PBIAS was percent (%).

R2 describes the proportion of the variance in the measured data explained by the model, ranging from 0 to 1, with higher values indicating less error variance. The NSE indicates the consistency between the plot of the observed vs. simulated data and the 1:1 line, ranging between -∞ and 1.0 (1 inclusive) and optimized at the value of 1. The PBIAS measures the average tendency of the simulated data to be larger or smaller than the observation, and a lower value implies a more accurate model simulation, with optimization at 0.0. Positive values indicate a model underestimation bias, and negative values indicate a model overestimation bias. Moriasi et al. (2007) proposed a widely used rating system, which judged the modelling performance as “very good”, “good”, “satisfactory”, or “unsatisfactory” using PBIAS < ±10 %, ±10 % ≤ PBIAS < ±15 %, ±15 % ≤ PBIAS < ±25 %, or PBIAS ≥ ±25 % respectively or using 0.75 < NSE ≤ 1.0, 0.65 < NSE ≤ 0.75, 0, 0.50 < NSE ≤ 0.65, or NSE ≤ 0.50 respectively.

Simulations forced by the long-term climatic data series revealed root development using the static and dynamic rooting depth approaches (Fig. 8a). Instead of assigning a fixed root depth of 5.0 m in the static rooting depth approach, the dynamic rooting depth approach simulated the root depth extension, which is consistent with the data from the literature. It was found that the rooting depth may be as deep as 11.0 m below the ground. The simulated rooting depth extension rate slowed as the stand age increased, which was also consistent with the observations in artificial forests (see in Fig. S4; Christina et al., 2011; Wang et al., 2015). Furthermore, the simulated root profiles by the two approaches within a 2.0 m soil depth were also evaluated against the data collected from the literature (Fig. 8b, c). The simulated results varied within the range provided by the data in the literature. Regressions between the mean fine-root fractions by the static and dynamic rooting depth approaches and that of the literature data provided R2 values of 0.34 and 0.64 respectively, indicating that the dynamic approach performed better than the static approach.

The evolution of root growth and soil water over the long term is depicted in Fig. 9. Visually, wetting and drying processes over the soil profiles were commonly found from simulations of both root growth modelling approaches. However, the difference in the simulated soil water distribution was also significant (Fig. 9d).

These two approaches resulted in substantially different spatial distributions of roots and soil water and, thus, a significant difference in root–water interactions. The soil water varied significantly with precipitation during the entire period. In most years, the maximum infiltration depth was less than 2.0 m. Meanwhile, precipitation could infiltrate down to 5.0 m in consecutive wet years, for example, the period from the 10th to 15th years. Notably, a time lag effect existed between the peak precipitation and the maximum infiltration depth: the peak precipitation occurred around August, while the maximum infiltration depth was reached around March in the following year.

Precipitation replenishes soil water through infiltration. The amount and depth of the infiltrated water impact the growth and water use of the root systems. The infiltration is associated with the a priori soil water content and its distribution over the profile, the amount and duration of an individual precipitation event, and the effects of the randomly sequential events. Establishing a relationship between the infiltration depth and precipitation on the basis of a single event is complicated and difficult to achieve. Instead, analyses were performed on an annual basis; that is, the maximum infiltration depth vs. the annual precipitation amount were regressed for the two root growth modelling approaches, as shown in Fig. 10. During the simulation period of 50 years, the annual precipitation varied from 250 to 850 mm. It was found that the annual infiltration amounts from these two approaches were exceptionally close to each other. The dynamic rooting depth approach was 4.0 % lower than that of the static rooting depth approach, which is an insignificant difference, as discussed in a later section. The maximum infiltration depth was positively correlated with the annual precipitation, which is not uncommon. The results also indicated that the maximum infiltration depth may reach 6.0 m below the ground in very wet years. Interestingly, the regression lines of these two root growth modelling approaches crossed at approximately 500 mm of annual precipitation. When annual precipitation was less than 500 mm, the infiltration reached deeper soil for the static rooting depth approach, and the inverse was found when the annual precipitation was more than 500 mm.

The fine-root distributions simulated by these two approaches showed exceptionally similar patterns in the soil profile (Fig. 11a). However, a quantitative difference between them was also noticeable. For the static rooting depth approach, roots grow only within the soil layer with a present 5.0 m thickness. For the dynamic rooting depth approach, the coarse roots reach 11.0 m below the ground. Within the top 2.0 m soil layer, the fractions of fine roots for the static and dynamic rooting depth approaches were 90.0 % and 80.3 % respectively, and within the 2.0–5.0 m soil layer, these values were 10.0 % and 14.7 % respectively. For the dynamic rooting depth approach, only 5.0 % of fine roots were in the soil below 5.0 m.

The distribution of root water uptake over the soil profile was similar to that of the fine-root density (Fig. 11b). The static rooting depth approach resulted in an annual water uptake of 338 mm (SD = 84 mm), whereas the dynamic rooting depth approached 381 mm (SD = 84 mm). In the top 2.0 m soil layer, the root uptake was 298 and 318 mm for the static and dynamic rooting depth approaches respectively. In the 2.0–5.0 m layer, the values for the static and dynamic rooting depth approaches were 40 and 40 mm respectively. Below 5.0 m, the dynamic approach resulted in a root uptake of 24 mm, which accounted for 6.2 % of the uptake from the entire profile.

The long-term soil water evolution is illustrated in Fig. 9. Vertically, the most active soil water zone was within the top 2.0 m (statistically based on the simulation results). This was confirmed by root development and water uptake in this study (Fig. 11) and by field observations in this region (Suo et al., 2020). In the top 2.0 m soil layer, infiltration events replenish and plant roots deplete the soil water alternately. The two root growth modelling approaches resulted in remarkably similar soil water enrichment and depletion patterns in the top 2.0 m layer, especially during the infiltration period (Fig. 9d). Below the top 2.0 m, soil water was continuously in a negative state, and the dynamic rooting depth approach resulted in more soil water depletion than the static rooting depth approach.

Drying soil layers have been frequently reported in previous studies in the Loess Plateau (Y. Wang et al., 2018). The upper and lower boundaries of the drying soil layers were defined according to the soil desiccation index (Wang et al., 2011), as shown in Fig. 12. The static rooting depth approach did not demonstrate the sustainable existence of drying soil layers. This is because it pre-sets a fixed rooting depth over the simulation period, which does not capture the hydraulic traits of roots that may develop to use water from the wetter zone beneath the pre-set rooting depth.

For the dynamic rooting depth approach, the drying soil layers started to develop at a stand age of approximately 8 years and became deeper and thicker sustainably with increasing stand age. Its lower boundary extended gradually to deeper soil, while its upper boundary fluctuated significantly. The strong fluctuations were associated with the infiltration, whereas the continuous extension of the lower boundary was due to sustained plant root uptake and rare recharge. With stand ageing, the soil water content in the drying soil layer decreased continuously, and the thickness increased gradually.

The simulated evolution of the drying soil layers was partially confirmed by field observations (Jia et al., 2017a), as shown in Fig. 12. The simulation produced a progressive course of the lower boundary, which was far deeper below the ground than the observations. The observations were limited to the top 5.0 m only. A deeper sampling might have resulted in a thicker drying soil. In Changwu, which is 80 km from the study site in the Loess Plateau, it was reported that the thickness of the drying soil layer reached 7.0 m, with its lower boundary as deep as 8.0 m below the ground in the black locust plantation (Li et al., 2008). It should also be noted that the sampling depth was limited to 8.0 m. Deep sampling has indicated that drying soil layers reach a depth of 19.0 m in the Loess Plateau (Wu et al., 2021).