Hydrological Sciences—Journal—des Sciences Hydrologiques, 43(4) August 1998 Special issue: Monitoring and Modelling of Soil Moisture: Integration over Time and Space 549

Simulation of root development based on the dielectric breakdown model

J. CHIKUSHI Biotron Institute, Kyushu University, Fukuoka 8128581, Japan e-mail: chiku2@agr.kyushu-u.ac.jp

O. HIROTA Institute of Tropical Agriculture, Kyushu University, Fukuoka 8128581, Japan e-mail : hirotao@agr. kyushu-u. ac. jp

Abstract Knowing the root distribution in soil is essential for estimating water uptake by plant roots. It is difficult, however, to characterize and model undisturbed root systems. Root development in a two-dimensional potential field is simulated with the dielectric breakdown model (DBM), which implies a similarity between electric discharge and root distribution. A weighted potential gradient with an exponent rj was used to create probability distributions of root growth. Each root was generated step-by-step by selecting random numbers from probability distributions. Simulations were performed for point sources and horizontal line sources. In both cases, as n increased, the root system changed from a cluster and dense type to a slender and sparse type and the fractal dimension decreased. For a point source of water, there was a definite pattern of root development toward the source. It is concluded that the DBM effectively describes root development under various water conditions.

Simulation du développement racinaire par le modèle de rupture diélectrique Résumé La connaissance de la répartition des racines dans le sol est essentielle pour estimer l'absorption de l'eau par les plantes. Il est cependant difficile de caractériser et de modéliser systèmes racinaires non perturbés. Cette étude est consacrée à la simulation du développement des racines dans un champ de potentiel bidimensionnel en appliquant un modèle de rupture diélectrique (Dielectric Breakdown Model— DBM). L'emploi de ce modèle se justifie par l'analogie entre décharge électrique et répartition des racines. On a utilisé un gradient de potentiel pondér, par un exposant n pour créer des distributions de probabilité de croissance des racines. Les racines simulées ont été produites par itération en générant des nombres aléatoires suivant les distributions de probabilité spécifiées. Deux sources d'eau ont été étudiées: sources ponctuelles et sources linéaires. Dans les deux cas, les systèmes racinaires simulés passent d'une allure groupée et dense à une allure allongée et éparse et leur dimension fractale diminue lorsque la valeur de n augmente. A chaque type de sources correspond un mode précis de développement des racines. On peut conclure que le modèle de rupture diélectrique s'est révélé efficace pour décrire le développement des racines des plantes cultivées dans des régimes hydriques variés, en rendant possible la simulation de différents types de systèmes racinaires.

INTRODUCTION

Water uptake is very important for modelling water flow in a soil^plant-atmosphere continuum (SPAC). Water uptake models have been classified into microscopic and macroscopic types. Microscopic models consider water uptake as the convergent

Open for discussion until 1 February 1999

550 J. Chikushi & O. Hirota

radial flow towards and into a representative individual root taken to be a line or narrow-tube sink uniform along its length. Solutions of the microscopic model have been shown by Gardner (1960) and Taylor & Klepper (1975). These solutions however ignore sectional differences in root activity due to differences in age or location (Hillel, 1980). In macroscopic models, the root system is regarded as a diffuse sink that penetrates each soil layer uniformly, though not necessarily with a constant strength throughout the root zone. This approach was taken by many researchers, such as Gardner (1964), Whisler et al. (1968), Molz & Remson (1970, 1971), Nimah & Hanks (1973), Hillel et al. (1976), and Feddes et al. (1978). Macroscopic models do not take into account the increase of suction in the soil water close to the roots (Hillel, 1980). In most of these models, one-dimensional root density distributions have been considered.

Taylor & Klepper (1975) stressed the importance of spatial distribution of roots in the water and nutrient uptake. Lafolie et al. (1991) developed a two-dimensional model for soil water movement based on the observation of root cluster in a field (Tardieu, 1988). Using the model, Tardieu et al. (1992) found that soil resistance was up to two factors of ten higher in clumped root spatial arrangements compared to regular arrangements. This suggests that the root spatial arrangement could have a crucial effect on soil-plant water relationships. Arya et al. (1975) also indicated that the root density distribution, flow pattern and water uptake are nonuniform in the two-dimensional space.

Soil water uptake by plant roots is associated with temporal root behaviour as well as spatial root distribution. Roots grow in different directions and spacing, and at different rates. Taylor & Klepper (1975) suggested that the root distribution can change markedly within a period of several days. Hillel & Talpaz (1976) simulated water uptake for different root growth patterns resulting from several concurrent processes, including proliferation, extension and death. Protopapas & Bras (1987) also integrated plant growth in their model.

These ideas urged us to make more realistic SPAC models, involving physical, biological and hydrological processes of plant growth. Although several observation techniques have been developed, such as the mini-rhizotron (Taylor et al, 1970; Upchurch & Ritchie, 1983) and tomography (Anderson & Hopmans, 1994), much difficulty still exists in measuring root geometries. Thus, there is a need for a model that generates root growth, as a sub-model of SPAC models.

Some studies on root morphology have been conducted (Shibusawa, 1994; Pages & Kervella, 1990; Fitter, 1982; Lungley, 1973). However, most of them do not consider the environment around roots, especially water content. Hydrotropism, which is the property of roots to grow toward moisture, can be related to hydraulic potential gradients because plant roots seek to optimize their arrangement to extract soil water effectively from a nonuniform water content distribution.

Root patterns can be characterized by their fractal dimension (Tatsumi et al, 1989), and the dielectric breakdown model (DBM) can generate branching patterns for different fractal dimensions (Niemeyer et al, 1984). In the present study, the DBM was applied to the root development in a potential field, and the characteristics of the parameters involved in the model were examined.

Simulation of root development based on the dielectric breakdown model 551

MODEL DESCRIPTION

Lou & Kato (1987) reported that low soil moisture conditions promote root development, compared to moderate and wet conditions. However, in perfectly dry soil, roots cannot extend themselves and would wilt. The root growth is likely to follow the conditions of water uptake demand and the existence of appropriate water sources in the root neighbourhood. Thus, it is reasonable to hypothesize that roots grow optimally depending on hydraulic potential gradients.

Niemeyer et al. (1984) developed a simple stochastic model for the two-dimensional discharge due to dielectric breakdown. Discharge patterns obtained by the model have a fractal structure similar to root patterns. A two-dimensional system was also assumed for the root development simulation. Furthermore, a driving potential, (|>, including the contribution of water potential was considered. In the potential field, the DBM could be applied to the formation of a root system. Moreover, it was assumed the plants were dicotyledons without tillering. After a seed is placed on the soil surface, the first step of downward extension is generated. For each extension (or time) step, the potential field can be generally expressed by the Laplace equation:

V2c|) = 0 (1)

In a two-dimensional field (it is possible to treat three-dimensional cases), this equation can be approximated by the following discrete equation:

4>/j = 4(^-1 , / + * - ' , / +*',/-> + <K/+ i ) <2>

where (j>,; is the potential at the grid point (i, j). The Laplace equation can be solved by a relaxation method at each step. For all grid points established as roots, the boundary condition becomes

4>,v = 0 (3)

For points corresponding to the source (e.g. a water source):

f j = 1 (4)

and for boundaries that are impermeable for any substance:

d<|> / = 0 (5) an

where n is the coordinate normal to the impermeable boundary. As for the root extension, every grid point corresponding to the root has the

possibility to extend farther or to make a new branch. The possibility p that a root at point (i, j) will extend to a candidate point (ï, f) can be defined as (see Fig. 1):

552 J. Chikushi & O. Hirota

•• —O Candidate Fig. 1 Established and candidate points for root development.

where the parameter r\ determines the probability function. Root extensions or branches are determined at each step by selecting one of these candidates based on a generated random number. At first, the simulation domain is sectioned into grids. Roots are developed along grid sides, where the grid size can be expressed by a length scale without special unit. Thus, simulated root systems do not exhibit smooth curves and specific temporal variations, but only approximate real root patterns.

Next, the fractal dimension of the simulated root systems was estimated. This can be done by superimposing two-dimensional grids of varying size, r, on the root system, counting the number of interceptions, N, of each grid with the root system, and plotting logN against the grid r as follows (Tatsumi et al., 1989):

N{r)ccr~D (7)

The D value is determined as the slope of the regression line of ln/V(r) vs lnr.

SIMULATIONS

Simulation 1: extension toward a line source

The root development from a seed was simulated in a two-dimensional region. The initial potential field was assumed to be varying linearly with depth from 1.0 (line source) to 0.0 (soil surface). The line at the bottom of the region was also assumed to be a line source. Figure 2 shows the simulated results for different n values (il = 0.3, 0.5, 0.8, 1.0 and 2.0) by 1000 steps (except Fig. 2(e)—by 500 steps). The root pattern varied with r\ value. That is, for n = 0.3 a lumped root system was formed near the surface, whereas for r\ = 2.0 the tap root notably developed and the root system branched sparsely. Figure 2(f) shows the development at 500 steps following Fig. 2(e). After the root reaches the line source, lateral and even upward growth of branched roots occurs. Between r| = 0.3 and 2.0, the pattern of the root system changes markedly from a clumped, shallow and dense type to a sparse, deepened and thin type. The case of n =0 .8 shows a moderately balanced root system. The root systems of Fig. 2(a) and (d) closely resemble those of Anemone

Simulation of root development based on the dielectric breakdown model 553

Distance Distance Distance 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50

Q -I ! ' 1 1 (- -J •—I 1 1 I l~ -I 1 1 1 1

100 H 1 1 1 1 b H H *i H r* 4- H h ^ H r* 4-

(d) TI=1.0 (e) TI=2.0 (f) i1=2.0

500 steps 1000 steps Fig. 2 Root system patterns simulated under the condition of a bottom line source.

patens var. wolfgangiana and Gutierrezia diversifolia observed in the Great Plains of Canada (Coupland & Johnson, 1965), respectively,

Figure 3 shows examples of the potential distribution for r\ values of 0.3, 0.8 and 2.0. The highest potential gradient, i.e. the smallest distance between equi-potential line and root entity, occurs at the lowest part of the root systems (Fig. 3(a) and (b)). This means that positions closer to the source have a higher potential to extend than the upper positions left behind in the growth. After roots reach the source (bottom) as in Fig. 3(c), some lateral roots at the tip of the root system branch

554 J. Chikushi & O. Hirota

Distance Distance Distance 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50

=0.25

(a) n=0.3 (b) n=0.8 (c) n=2.o Fig. 3 Potential distribution under the condition of a bottom line source.

upward because the bottom line changes from source to root, as the potential changes from 1 to 0.

Table 1 shows the fractal dimension D for different r\ values. The D value declined with r\, i.e. from 1.7 (for r\ = 2.3) to 1.27 (for r\ = 0.3). Tatsumi et al.

10" -0.5

X 1.5 _ 0 0.5 1

Log (Grid size)

Fig. 4 Plotted data and the regression lines for the relationship between the intersected number of grid and the side length of grid.

Simulation of root development based on the dielectric breakdown model 555

(1989) indicated that the fractal dimension for common millet {Panicum Miliaceam L.) was about 1.48. This fractal dimension (D) is nearly equal to that for -n = 1.0. Actually, a similar pattern was observed for these root systems. Two examples of the relationship between N and r are shown in Fig. 4. The fit of equation (7) with the data is excellent.

Table 1 Fractal dimension for different values of r\.

11 0.3 0.5 0.8 1.0 2.0*

D 1.71 1.61 1.54 1.49 1.27

Correlation coefficient 0.999 0.998 0.998 0.996 0.995

* corresponding to Fig. 2(e).

Simulation 2: extension toward a point source

Figure 5 shows the results for r) = 0.3, 0.8 (1500 steps) and 1.3 (1000 steps), where we initially assumed a low potential (cj) = 0.001) for every grid point in the region except for a point source located at the bottom right corner of the region with high potential (c|> = 1.0). The tendency toward the point source was strengthened with increasing r\, though the tendency was scarce for the root system of r\ = 0.3. The corresponding potential distributions exhibit the tendency that roots extend toward a source for r\ = 0.8. The potential distributions of Fig. 5(a) and (b) differed the least. In Fig. 5(c), the highest potential occurred at the bottom left because the root had already reached the source.

Figure 6 shows the root development processes and their potential distributions for r| = 0.8. The developed roots closer to the source tend to be left behind in growth except at the front of the root system. The potential distribution near the source hardly changed during the process of root development.

Simulation 3: effect of random number range adopted

By generating uniform random numbers in the range of 0-1 one can achieve development of roots even at positions with low potential gradients. This leads to finely-branched and clumped root systems like that shown in Fig. 2(a). In the field, however, we sometimes observed a less branched root system near the stock base. To generate such a root system, we limited random numbers exceeding a critical value.

Figure 7 shows the results for r\ = 0.3 and 1.0, and for critical values of 0.5 or 0.8. In these cases, candidates below the critical value were completely excluded.

556 J. Chikushi & O. Hirola

Distance 20 40 60 100

Distance 20 40 60

(a) i i=0.3

Distance

(b) TI=0.8

100

(c) TI=1.3

Fig. 5 Root system patterns (solid lines) and potential distributions (dashed lines) simulated for a point source at the bottom right corner.

Thus, the upper part of the root system may be maintained with little growth, whereas the bottom of the rooting zone actively goes down. In particular, for the critical value of 0.8, a sparse root pattern was formed near the soil surface rather than in deeper parts (Fig. 7(b) and (d)). Similar patterns have been observed in the root system of a chrysanthemum plant variety (Psoralea lanvolata) (Coupland & Johnson, 1965).

DISCUSSION

The morphology of a root system is affected by two factors: genetics and environment (Smucker & Aiken, 1992). The L-system (being one of the pattern

558 ./. Chikushi & O. Hirota

Distance Distance Distance Distance 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50

0

Dep

th

10 -

20 -

30 -

40 -

50 -

60 -

70 -

80 -

90 -

100 -

(a) T]=0,3 (b) ii=0.3 (c) TI=1.0 (d) TI=1.0

Fig. 7 Change in root system pattern depending on the range of random number and r\ value. Critical values of the random number are 0.5 for (a) and (c), and 0.8 for (b) and (d).

high (Fig. 5(a) and (b)). This tendency may lead to the heterogeneous root density distribution in two-dimensional space as reported by Arya et al. (1975), and in turn could produce irregular patterns of water movement and root water uptake.

Plant roots have the functions of water and nutrient uptake and anchoring. To accomplish these functions, roots seem to optimize their performance. That is, roots develop to optimize environmental conditions such as soil water content, nutrient content, gravity, light, and soil strength. Of these, soil water content is the most important factor for the root development, though each factor has its own potential. In this study, we dealt only with water potential as a driving force for root growth. However, including various other environmental factors should be more practical. The DBM could be applicable also to such an extended potential field.

It is known that the tap root and first-order laterals contribute less to water uptake than second order laterals in soil (Smucker & Aiken, 1992). Secondary roots located between primary roots reduce soil moisture during the development of them­selves and higher-order roots. The soil moisture reduction makes it difficult for the higher order roots to grow. Thus, the distribution of second order laterals is very important for water uptake. Furthermore, roots finer than second-order laterals can invade smaller sized pores and can effectively absorb water in the space between relatively large-sized roots. However, high order laterals cannot develop excessively in the space between first-order laterals because of declining potential gradients due to the development of second-order laterals. The DBM can simulate these situations of actual root system reasonably well.

An alternative fractal growth model is the diffusion-limited aggregation (DLA) model (Witten & Sander, 1981). In this model, particles are added at random to sites at large distance (the source in the present study) from the seed and proceed ran­domly until they reach a site adjacent to the cluster (roots in the present study).

Simulation of root development based on the dielectric breakdown model 559

When this happens, the site is joined to the cluster. In the DLA, the Laplace equation (equation (6)) is also satisfied (Vicsec, 1989). However, concerning root development, the DBM may be preferable to the DLA because the DBM can be applied even in a heterogeneous potential field.

This root development model (DBM) still needs to be combined with water movement in soil. Roots are usually exposed to infiltrating water and thus varying soil water potential conditions. Thus, root growth is affected by elapsed exposure as well as instantaneous exposure to the soil water potential. Furthermore, the alloca­tion of photo-assimilates to the root is involved in root growth, along with water and nutrients absorbed from the soil. Such a dynamic crop growth model was already presented by Protopapas & Bras (1987). In this model, the root biomass is allocated to distinct layers in the root zone. In addition, roots were divided into young and old ones, and water uptake rate was evaluated according to root activity and root biomass. In the future, this idea can possibly be incorporated into the DBM. Then, allometry (France & Thornley, 1984) may play a role in estimating the amount of root biomass.

In conclusion, the dielectric breakdown model which was used to simulate root development could be useful to generate very realistic root systems characterized by genetic and environmental properties. Hopefully, this type of model will be inte­grated with SPAC models to describe more realistically water flow from soil to atmosphere through plants.

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