Plant and Soil 252: 251–265, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

251

Growth of a root system described as diffusion. II. Numerical model andapplication

Marius Heinen1,3, Alain Mollier2 & Peter De Willigen1

1 Alterra, PO BOX 47, NL-6700 AA Wageningen, The Netherlands. 2 INRA, Unite d’Agronomie, 71, AvenueEdouard-Bourlaux, B.P. 81, 33883 Villenave d’Ornon Cedex, France. 3Corresponding author∗

Received 3 June 2002. Accepted in revised form 19 November 2002

Key words: eastern white cedar, gladiolus, modelling, maize, numerical solution, rockwool, root length density,tomato

Abstract

In simulation models for water movement and nutrient transport, uptake of water and nutrients by roots formsan essential part. As roots are spatially distributed, prediction of root growth and root distribution is crucial formodelling water and nutrient uptake. In a preceding paper, De Willigen et al. (2002; Plant and Soil 240, 225–234) presented an analytical solution for describing root length density distribution as a diffusion-type process. Inthe current paper, we present a numerical model that does the same, but which is more flexible with respect towhere root input can occur. We show that the diffusion-type root growth model can describe well observed rootingpatterns. We used rooting patterns for different types of crops: maize, gladiolus, eastern white cedar, and tomato.For maize, we used data for two different types of fertiliser application: broadcast and row application. In case ofrow application, roots extend more vertically than horizontally with respect to the broadcast application situation.This is reflected in a larger ratio of diffusion coefficients in vertical versus horizontal direction. For tomato, weconsidered tomatoes grown on an artificial rooting medium, i.e. rockwool. We have shown that, in principle,the model can be extended by including reduction functions on the diffusion coefficient in order to account forenvironmental conditions.

Introduction

In a preceding paper, De Willigen et al. (2002) presen-ted an analytical solution for describing root growthas a diffusion process. Analytical solutions have theadvantage of easily exploring the behaviour of the pro-cess and examining limiting situations. However, toobtain analytical solutions, usually all sorts of sim-plifications have to be applied. Numerical methodsare more flexible, in the sense that more complicatedsituations can be considered. Therefore, we presenthere a numerical solution for root growth describedas a diffusion process, which can be incorporated in

∗ FAX No: +31(0)317-419000.E-mail: m.heinen@alterra.wag-ur.nl

numerical simulation models for describing root up-take of water and nutrients as a function of root lengthdensity distribution. In the numerical solution, we willincorporate the possibility of allowing the diffusioncoefficients be functions of environmental conditions,and the possibility that input of roots can occur at anyposition.

As the numerical root growth model has been used(in other studies) in combination with an existing two-dimensional simulation model (FUSSIM2) – both incartesian and cylindrical co-ordinates – for water flow,solute transport, and root uptake of water and nutri-ents (FUSSIM2; Heinen and De Willigen, 1998, 2001;Heinen, 2001), we focus on two dimensions (2D),either in cartesian or cylindrical co-ordinates.

252

Table 1. Explanation and units of the main symbols used in the text

Symbol Explanation Units

A total geometry coefficient of central node (I,J) cm3 d−1

A coefficient matrix consisting of entries AE and AS cm3 d−1

AC(entral) geometry coefficient of central node (I,J) cm3 d−1

AE(ast) D-geometry coefficient between central node (I,J) and its cm3 d−1

eastern neighbour (I+1,J)

AN(orth) D-geometry coefficient between central node (I,J) and its cm3 d−1

northern neighbour (I,J-1)

AS(outh) D-geometry coefficient between central node (I,J) and its cm3 d−1

southern neighbour (I,J+1)

AW(est) D-geometry coefficient between central node (I,J) and cm3 d−1

its western neighbour (I-1,J)

B known quantities in difference equation cm d−1

B vector with the known values of B cm d−1

C root length density cm (root) cm−3 (soil)

C0 part of the analytical solution of De Willigen et al. (2002) in case cm (root) cm−3 (soil)

of diffusion in the vertical direction only

Cn part of the analytical solution of De Willigen et al. (2002) cm (root) cm−3 (soil)

that accounts for diffusion in both directions

Cref some reference root length density, taken equal to 1 cm (root) cm−3 (soil)

Ctot total root length in region 0<X<L and 0<Z<ZL cm

C vector with the unknown C at the next time step cm cm−3

CV control volume dimensionless

DR radial root growth diffusion coefficient cm2 d−1

DX horizontal root growth diffusion coefficient cm2 d−1

DZ vertical root growth diffusion coefficient cm2 d−1

I column number of CV (used as subscript) dimensionless

J row number of CV (used as subscript) dimensionless

L half the distance between two plants cm

M number of rows dimensionless

N number of columns dimensionless

Nobs number of observations dimensionless

O observed root length density cm (root) cm−3 (soil)

Q root length density input rate cm (root) cm−3 (soil) d−1

QL root length growth rate cm cm−2 (soil) d−1

QM root biomass growth rate g cm−2 (soil) d−1

R radial co-ordinate cm

R0 average root radius cm (root)

S simulated root length density cm (root) cm−3 (soil)

T time d

Tf age of root system d

X horizontal co-ordinate cm

X1 half the row width, i.e. half the window of input of roots cm

Y third co-ordinate in cartesian co-ordinate system cm

Z vertical co-ordinate cm

ZL depth of the region for which Ctot is determined cm

d dry matter content of a root g (dry) g−1 (fresh)

fI,J indicator if roots enter the CV (fI,J = 1) or not (fI,J = 0) dimensionless

fh(h) reduction function for pressure head h as shown in Figure 2 dimensionless

253

Table 1. Continued

Symbol Explanation Units

h pressure head cm

hA some value of h for which fh = 1 as used in last example cm

hB some value of h for which fh = 0.065 as used in last example cm

i index referring to CV interface between columns I and (I+1) (used as subscript) dimensionless

j index referring to CV interface between rows J and (J+1) (used as subscript) dimensionless

n integer counter dimensionless

q geometrical conversion factor: ratio of area occupied by the plant to the cm−1

total volume of all CVs where root input occurs (Equation 14)

ssq sum of squared differences between observed and simulated (cm (root) cm−3 (soil))2

root length densities

�RI thickness of cylinder I (cylindrical co-ordinate system) cm

�T time step d

�Tm maximum �T in numerical computations d

�XI thickness of column I cm

�ZJ thickness of row J cm

� specific root decay rate d−1

αn parameter defined by Equation (19) cm−1

δRi radial distance between node (I,J) and (I+1,J) (cylindrical co- cm

ordinate system)

δXi horizontal distance between node (I,J) and (I+1,J) cm

δZj vertical distance between node (I,J) and (I,J+1) cm

ϕ angular co-ordinate (cylindrical co-ordinate system) cm

κ counter dimensionless

ρ density of a root g (fresh) cm−3 (root)

Materials and methods

Diffusion equation

In the preceding paper, De Willigen et al. (2002),following Hayhoe (1980) and Acock and Pachepsky(1996), described root growth by a diffusion equation.Root decay was incorporated as a first order sink term,while root input was treated as a boundary condition.To increase the flexibility for root input to occur atany position, we use here an extended diffusion equa-tion with root growth treated as a source term. Thegoverning diffusion equation for root length density in2D cartesian co-ordinates – with no gradient in the Ydirection – then reads (cf. De Willigen et al., 2002)

∂C

∂T= ∂

∂X

(DX

∂C

∂X

)+ ∂

∂Z

(DZ

∂C

∂Z

)− �C + Q

(1)

where C is the root length density (cm cm−3), T is time(d), X is the horizontal co-ordinate (cm), Z is the ver-

tical co-ordinate (cm), DX is the diffusion coefficientin the X direction (cm2 d−1), DZ is the diffusion coef-ficient in the Z direction (cm2 d−1), � is the specificroot decay rate (d−1), and Q is the root length densityinput rate (cm cm−3 d−1). So far, no physiologicalmeaning can be given to the diffusion coefficients. Allsymbols used in this paper are explained in Table 1;De Willigen et al. (2002) used capital symbols for vari-ables with a dimension, which is used here as well.The variables C, DX, DZ, and Q are functions ofspace (X, Z). C and Q are functions of time T. Thevariables DX, DZ , � and Q may also be a functionof conditions (see section ‘Root growth influenced byexternal factors’ for an example). Root developmentin the third dimension Y does not occur by assuming∂C/∂Y = 0.

In cylindrical co-ordinates – with no gradients inthe angular direction ϕ – the governing equation reads

254

(cf. De Willigen et al. 2002)

R∂C

∂T= ∂

∂R

(RDR

∂C

∂R

)

+R∂

∂Z

(DZ

∂C

∂Z

)− R�C + RQ (2)

where R is the radial co-ordinate (cm) and DR is thediffusion coefficient in the R direction (cm2 d−1).

Boundary conditions

The conditions at the boundaries must be known tosolve Equations (1) and (2). Because in Equations (1)and (2) the input of roots into the soil is modelled as asource Q (see section ‘On root input parameter Q’ forfurther discussion on Q), the input is not assigned to aboundary condition. In what follows we will assign allboundaries as no-flow boundaries: no roots will passthe air-soil interface, vertical boundaries have to bechosen such that they are lines of symmetry perpen-dicular to the soil surface, and the bottom boundary ischosen (far) below the extent of the root zone.

Numerical solution

Control volume finite elementsThe diffusion equation is numerically solved us-ing the control-volume-finite-element method (Heinenand De Willigen, 1988, 2001; Patankar, 1980). Theroot zone is divided into so-called control volumes.Figure 1 shows schematically how a root zone canbe divided into control volumes (CV); it can be ap-plied both for 2D cartesian and 2D cylindrical co-ordinate systems. In Figure 1, the geometric notationconvention used in this paper is given as well. Inthe two-dimensional cylindrical situation there is nogradient in the angular direction. The control volumesin this case are circular. In a vertical plane, the controlvolumes are seen as rectangles, like in Figure 1. So,Figure 1 can also be use for the cylindrical situation,in which case the X co-ordinate is replaced by the R co-ordinate, and proper volumes are used. Each controlvolume is represented by a node which is referred toby its column number I and row number J. In what fol-lows, I and J are used as subscripts referring to a CV.The small indices i and j refer to the interfaces betweentwo CVs. Except for the nodes of CVs at the boundar-ies, the nodes lie in the centre of the CVs. The reasonfor having the nodes locate at the boundaries is forproper treatment of flux boundary conditions (Heinen

Figure 1. (A) Example of node-centred control volumes with ‘half’control volumes at the boundaries; the hatched area in (A) is en-larged in (B) showing the Control Volume (CV) (shaded area)around node (I,J) including notation conventions.

and De Willigen, 1998, 2001; Patankar, 1980). In totalthere are N columns with CVs and M rows.

Numerical integrationEquations (1) and (2) are integrated for each CV overspace and time according to:

0 =∫ Xi,j

Xi−1,j

∫ Y+�Y

Y

∫ Zi,j

Zi,j−1

∫ T +�T

T

[−∂C

∂T

+ ∂

∂X

(DX

∂C

∂X

)+ ∂

∂Z

(DZ

∂C

∂Z

)

− �C + Q

]dT dZ dY dX

(3)

and:

0 =∫ Ri,j

Ri−1,j

∫ 2π

0

∫ Zi,j

Zi,j−1

∫ T +�T

T

[−R

∂C

∂t

+ ∂

∂R

(RDR

∂C

∂R

)+ R

∂

∂Z

(DZ

∂C

∂Z

)

− R�C + RQ

]dT dZ dϕ dR

(4)

255

For convenience, the third co-ordinate (Y, with �Y =1, and ϕ and �ϕ = 2π) is also considered in theabove notation in order to obtain expressions in volu-metric units. Integration of Equations (3) and (4) israther straightforward (Appendix A), in which the fol-lowing assumptions were used. The value for DX,DR, or DZ at an interface between two CVs is com-puted as the geometric average of the two diffusioncoefficients belonging to these CVs. By taking thegeometric average, the smallest of the two diffusioncoefficients will determine the average more than byapplying arithmetic averaging. This will result in lessroot growth towards regions with smaller diffusioncoefficients. Secondly, the gradient of C at an interfacebetween two CVs is temporarily linearly approxim-ated. Thirdly, the decay of roots is based on the ‘old’value of C. Then, for each CV the following algebraicequation results:

AI,J CT +�TI,J + AW,I,J CT +�T

I−1,J + AE,I,J CT +�TI+1,J

+AN,I,J CT +�TI,J−1 + AS,I,J CT +�T

I,J+1 = BI,J (5)

where �T represents the time step (d). The coefficientA (cm3 d−1) is given by:

AI,J = AW,I,J + AE,I,J + AN,I,J + AS,I,J + AC,I,J

(6)

In cartesian co-ordinates the coefficients A (cm3 d−1)are given as:

AE(ast),I,J = DX,i,j�Y�ZJ

δXi

AW(est),I,J = DX,i−1,j�Y�ZJ

δXi−1

AS(outh),I,J = DZ,i,j�Y�XI

δZj

AN(orth),I,J = DZ,i,j−1�Y�XI

δZj−1

AC(entral),I,J = �XI�Y�ZJ

�T(7)

In cylindrical co-ordinates the coefficients A (cm3

d−1) are given as:

AE(ast),I,J = DR,i,jRi2π�ZJ

δRi

AW(est),I,J = DR,i−1,jRi−12π�ZJ

δRi−1

AS(outh),I,J = DZ,i,j

(R2

i+1 − R2i

)2π

2δZj

AN(orth),I,J = DZ,i,j−1

(R2

i+1 − R2i

)2π

2δZj−1

AC(entral),I,J =(R2

i − R2i−1

)2π�ZJ

2�T(8)

In cartesian and cylindrical co-ordinates the coefficientB contains known quantities (cm d−1) and is given by:

BI,J = AC,I,JCTI,J(1 − ��T) + AC,I,JQI,J�T (9)

Since flow across an interface between two CVs ap-pears in the expression for both CVs we have AW,I,J

= AE,I−1,J and AN,I,J = AS,I,J−1; this results in lesscomputer storage.

Setting up Equation (5) for all CVs we obtain a setof N×M equations with N×M unknowns, which canbe written in matrix notation as:

AC = B (10)

Figure 2 shows the structure of Equation (10) andspecifically the structure of coefficient matrix A con-taining the coefficients A. The vector C contains theunknown values of C for all CVs at time T+�T, andthe vector B contains the known variables B. Theformal solution of Equation (10) is given by:

C = A−1B (11)

where A−1 is the inverse of matrix A. Since the in-verse of matrix A is not easily obtained, the solutionof Equation (10) is not directly available. Since A issparse, we have chosen a convenient algorithm fromthe IMSL library (Visual Numerics Inc., 1997; usingthe method of Cholesky factorisation).

On root input parameter Q

In most cases, root growth is known as an increasein root biomass rather than an increase in root length.Root biomass input QM (g (root) cm−2 (soil) d−1),is easily converted into root length input QL (cm(root) cm−2 (soil) d−1) from:

QL = QM

πR20ρd

(12)

The denominator in Equation (12) represents theinverse of the specific root-length:mass ratio (cm(root) g−1 (dry matter root)). For instance, for a rootwith radius R0 = 0.015 cm, a density of ρ = 1.0 g cm−3

and a dry matter content of d = 0.08 g g−1, the specific

256

Figure 2. Schematic representation of the matrix notation Equation (10) showing the five banded sparsity pattern of the coefficient matrix Awith corresponding arrangement of the unknowns C and knowns B in vectors C and B, respectively.

root-length:mass ratio equals 17 684 cm (root) g−1

(dry root). For example, De Willigen and Van Noord-wijk (1987) presented data from the literature rangingfrom 800 cm (root) g−1 (dry root) (for a maize crop)to 72 300 cm (root) g−1 (dry root) (for a grass), withan average value of around 20 000 cm (root) g−1 (dryroot).

However, in Equation (1), root input must beexpressed per unit volume of soil where root inputoccurs. QI,J can be computed from QL according to:

QI,J = QLqfI,J, (13)

where f is an indicator which equals one if in that CVroots enter the soil or which equals zero if in that CVno roots enter the soil. So, f = 1 for the CVs wherethe root-shoot interface is located. Parameter q takescare of the conversion of input per unit area to inputper unit volume. If root input is confined to a volumeof soil with a certain depth interval �Z, then q is givenby:

q = L

X1�Z, (14)

where L is half the distance between two plants(cm) and X1 is half the row width (cm) (L and X1 areshown in Figure 1 of De Willigen et al., 2002). In whatfollows, we express root length input per unit of areaQL.

Parameter optimisation

The four parameters in Equation (1) or (2), i.e. DX orDR, DZ , � and Q, can be optimised against observed

root distributions, provided these are known at least attwo points in time. The parameters are optimised bysolving Equation (1) or (2) for the time interval T = 0to T = Tf . This was done using the Amoeba procedure(Press et al., 1996) plus, if desired, the Powell pro-cedure (Press et al., 1996). The minimisation criterionwas chosen to be:

ssq =Nobs∑κ=1

(Oκ − Sκ)2 (15)

where ssq is the sum of squared differences((cm cm−3)2), O is the observed root length dens-ity (cm cm−3), S is the fitted root length density(cm cm−3), and Nobs is the number of observations.The optimisation program was made flexible so thatthe lay-out of CVs (grid) did not have to be equal to themeasurement positions. In most cases C is measuredeither using pinboards or in core samples (Do Rosárioet al., 2000) in a regular grid. For numerical reasons, afiner grid for the simulation model may be required.In the latter case, Sκ for the same positions as themeasurement positions was obtained via double linearinterpolation in the two dimensions.

To prevent negative guesses for the parameters,the optimisation was carried out using log-transformedparameters. The actual predictions of C (= S) weredone with the ‘real’ values.

The optimisation program can also be used in thecase where one or more of the four parameters needs tobe fixed at a known value. For example, if informationis known about the constant input rate of root biomass,then Q can be fixed and only three parameters need tobe optimised.

257

The optimisation program is available from thefirst author by sending an E-mail request.

Verification

The numerical solution discussed above was verifiedagainst the analytical solution as presented in thefirst article of this series (De Willigen et al., 2002).It pertains to the situation of constant root input Q(Equations (26), (32) and (33) of De Willigen et al.,2002)

C = C0

π+ 2

π

∞∑n=1

Cn cos[αnX], (16)

where C0 is the part of the analytical solution in caseof diffusion in the vertical direction only and Cn isthe part of the analytical solution that accounts fordiffusion in both directions. C0 and Cn are given by:

C0 = πQL

2√

Dz� exp

[−Z

√�DZ

](1 + erf

[ −Z

2√

DZT+ √

T �])

−

exp[Z

√�DZ

] (1 − erf

[Z

2√

DZT+ √

T �])

(17)

and:

Cn = πQL

2√

DZ(� + α2nDx)

sin[αnX1]αnX1

exp

[−Z

√�+α2

nDX

DZ

] (1 + erf

[ −Z

2√

DZT

+√T (� + α2

nDX)])

−

exp

[Z

√�+α2

nDX

DZ

] (1 − erf

[Z

2√

DZT

+√T (� + α2

nDX)])

(18)

and with αn by

αn = nπ

L(19)

We consider a good correspondence between analyt-ical and numerical solution as a technical justificationof the numerical solution procedure.

Verification was carried out for the following set of(arbitrarily chosen) fixed parameters: L = 40 cm, X1 =5 cm, DX = 5.6 cm2 d−1, DZ = 56 cm2 d−1, � = 0.02d−1, QL = 3 cm cm−2 d−1. Results for the transientsituations T = 30 d and T = 100 d will be given forboth the numerical and analytical computations; at T =100 d a near steady state root distribution is obtained.Time steps in the numerical computations started at�T = 0.001 d which was successively increased by afactor 1.1, with a maximum time step of �Tm = 0.1 d.Non equal-sized CVs were used: small CVs were usednear the root input window. The successive columnwidths increased approximately logarithmically: �X= 0.1, 0.14, 0.2, 0.28, 0.39, 0.55, 0.77, 1.08, 1.5, 2.12,2.98, 4.18, 5.88, 8.25, 11.58 cm, respectively. For �Zthe same sequence was used followed by 14 additionallayers of 10 cm each. In Equation (16), 500 summa-tions were used to obtain the analytical solution. Inthe analytical computations, root input occurred in thefirst nine CVs of the first layer, i.e., 5 cm by 0.1 cm.In the analytical solution, the input window was X1 =5 cm.

Additionally, the analytical distribution was usedas input for the fitting procedure to determine if theoriginal parameters can be well estimated. This wasdone for both T = 30 d and T = 100 d. The initialguesses were DX = 5 cm2 d−1, DZ = 60 cm2 d−1,� = 0.03 d−1, and the root input rate was fixed at QL

= 3 cm cm−2 d−1. QL was fixed to demonstrate thatnot all four parameters need to be fitted. For example,information on root production may be known, for ex-ample from a plant growth model, and thus can be keptat that value.

Fitting to observed root distributions

To show the applicability of the diffusive root growthmodel, we investigated if the model can be fitted toobserved root distributions. For this purpose we usedonly a few extreme examples, rather than a large set ofexamples. The four examples are: (1) maize (Zea maisL. cv. Mandigo) for both broadcast and row fertilisa-tion (Schröder et al., 1996, 1997), (2) an eastern whitecedar (Thuja occidentalis L. ‘Brabant’; Pronk et al.,2002), (3) a bulb crop (Gladiolus grandiflorus Hort.,cv. Traderhorn), and (4) tomato (Lycopersicon escu-lentum L.) grown in rockwool. For example (4), twodata sets were used differing in cultivation manage-ment and cultivar: cv. Sonata (Van Noordwijk, 1978),and cv. Aromata. Example 1 is meant to determineif root growth differs for different types of fertiliser

258

Figure 3. Example of reduction function fh as a function of (abso-lute) pressure head h characterised by typical pressure heads h1, h2,h3, and h4; for example, h1 = 0 cm, h2 = 1 cm, h3 = 700 cm, and h4= 16 000 cm.

application and how this affects the values of the para-meters. Example (2) refers to a cylindrical co-ordinatesystem. Example (3) refers to a system where rootsenter the soil at a certain depth. Since the analyticalsolution can only consider input at the soil surface,this example shows the merit of a numerical model.Example (4) refers to an extreme situation where rootscan grow only in a limited rooting environment. Arockwool growth system consists of a rockwool slabof typical dimensions length 100 cm, width 15 cm andheight 7.5 cm, on top of which two to four small rock-wool cubes are placed (length 10 cm, width 10 cm,height 6.5 cm). Plants are planted in the rockwoolcubes.

Table 2 summarises some basic data for each ofthe examples, including the initial guesses of the para-meters to be fitted; the parameters were not restrictedwithin a range of allowable values. The observationswere obtained from a pinboard with a mesh sizeof 5×5 cm2. Except for eastern white cedar, in allcases the following standard values were used: R0 =0.01 cm, ρ = 1.0 g cm−3, and d = 0.05 g g−1; foreastern white cedar, a woody species, we used R0 =0.03 cm, ρ = 1.0 g cm−3, and d = 0.48 g g−1. Ex-cept for eastern white cedar and tomato, initially therewere no roots present. Eastern white cedar and tomatoare normally pre-grown in small planting cubes beforethey are transplanted into the soil or onto the rock-wool, respectively. For these two cases, the initiallymeasured C was used as starting root distribution.

Figure 4. Comparison between analytical (left column) and numer-ical (right column) root length density distributions at times 30 d(top row) and 100 d (bottom row). Each contour line represents aninterval in root length density of 0.5 cm cm−3, for L = 40 cm, X1 =5 cm (denoted by the black rectangle at the top of each plot), Dx =5.6 cm2 d−1, Dz = 56 cm2 d−1, � = 0.02 d−1, QL = 3 cm cm−2

d−1.

Root growth influenced by external factors

It is generally known that root functioning in ex-tremely wet or extremely dry regions is poor. One mayexpect that roots will not grow towards such regions.Here we propose to model this by making the diffusioncoefficient D a function of the pressure head h (cm)status of the soil:

D = fh(h) (20)

where the function fh is shown in Figure 3. At thistime, no data for the function fh are available. We pro-pose to use a similar function to that used for transpir-ation reduction by, e.g., Wesseling (1991); Figure 3shows fh(h) corresponding to the characteristic tran-

259

Table 2. Basic information of the examples: example ID, sampling time Tf (d), total sampling depth ZL (cm), total sampling width L (cm),

total number of sampling grids of 5×5 cm2, number (#) of sampling grids where roots were present (C > 0 cm cm−3), assumed location ofroot input (cm), number (#) of CVs used in the numerical simulation, and the initial guesses of the parameters to be fitted DX (or DR in case ofeastern white cedar) (cm2 d−1), DZ (cm2 d−1), QL (cm cm−2 d−1) and � (d−1)

Example ID Tf Depth,Width, Grids # grids Root input # CVs DX DZ QL �

ZL (cm) L (cm) with locationa (or DR)C > 0

Maize-row 47 40 35 56 42 0 < Z< 5 56 4 8.4 0.6 0.001Maize-broadcast 47 40 35 56 33 5 < Z< 10 56 4 3.7 0.5 0.005Cedar 215 25 35 35 24 5 < Z< 10 35 3 (DR) 3 1 0.02Gladiolus 72 40 55 88 23 10 < Z< 15 88 4.5 3 1.8 0.09Tomato, cv. Sonata 300 14 25 10 10 5 < Z< 6.25 100b 10 1 10 0.02Tomato, cv. Aromata 56 14 12 11 11 4.875 < Z< 6.5 60c 10 1 10 0.02

a Except for Tomato cv. Aromata, in all cases root input occurred for 0 < X < 5 cm; for Tomato cv. Aromata this was 0 < X < 2 cmb Of which 32 were not considered (no rooting medium present)c Of which 16 were not considered (no rooting medium present)

Table 3. Fitted values for DX (or DR in case of eastern white cedar) (cm2 d−1),DZ (cm2 d−1), � (d−1), QL (cm cm−2 d−1), and the sum of squared differencesssq ((cm cm−3)2) for the examples considered

Example ID DX (or DR) DZ QL � ssq

Maize-row 2.234 4.552 0.158 0.013 0.469Maize-broadcast 2.943 2.270 0.459 0.080 0.534Cedar 1.404 (DR) 9.802 0.597 0.597 5.740Gladiolus 3.011 1.354 2.641 0.192 3.764Tomato, cv. Sonata 13.865 0.233 13.011 0.046 202.064Tomato, cv. Aromata 12.914 2.935 22.189 0.135 0.486

spiration reduction pressure heads for wheat. Underextremely wet conditions oxygen will be limiting forroot functioning so that we assume that roots will notgrow towards saturated regions. Under extremely dryconditions, water will be limiting for uptake so thatwe assume that roots will not grow towards these dryregions. We present an example to mimic the effect offunction fh by letting roots grow in a soil with uniformhA for which fh(hA) = 1 (Figure 3) or in a soil withtwo regions differing in h: in the region 0<X<20 cm h= hA for all Z, and in the region 20<X<40 cm h = hB

for all Z for which fh(hB) = 0.065 (Figure 3). In bothcases, the root growth parameters are the same: DX =10 cm2 d−1, DZ = 5 cm2 d−1, QL = 5 cm cm−2 d−1,� = 0.02 d−1.

Similarly, root growth may be hindered by high saltconcentrations or low oxygen concentrations, whichcan be described by analogous functions.

Results and discussion

Verification

Figure 4 presents a contour plot view of the compar-ison between analytical (left column) and simulated(right column) C distribution for times T = 30 d (toprow) and T = 100 d (bottom row). The comparisonis very good. Note that the difference in consider-ing root input, (analytical solution: across the surface0<X<5 cm; numerical solution: in CVs in region0<X<5 cm and 0<Z<0.1 cm) does not result in a dif-ference between simulated C. The total root length inthe region 0<X<40 cm and 0<Z<180 cm was in bothcases almost identical: analytical (see Appendix B forderivation): 2707 cm (T = 30 d), 5117 cm (T = 100d), and numerical: 2709 cm (T = 30 d), 5190 cm (T= 100 d). The numerical total root length in the region0<X<40 cm and 0<Z<180 cm is always larger thanthe analytical, as in the numerical simulation all rootsare bounded in the simulated region, while the analyt-

260

ical solution extends to Z→ ∞ thus yielding also rootsbelow depth of 180 cm. Analytical Ctot for Z → ∞ is,respectively, 2707 and 5188 cm.

From the fitting procedure, the parameters DX, DZ

and � were estimated as DX = 5.66 cm2 d−1, DZ =56.73 cm2 d−1, and � = 0.0208 d−1 for the analyticaldata at T = 30 d. These are very close to exact values:DX = 5.6 cm2 d−1, DZ = 56 cm2 d−1, � = 0.02 d−1.For T = 100 d the following estimates are obtained:DX = 5.66 cm2 d−1, DZ = 56.72 cm2 d−1, and � =0.0206 d−1, which are identical to those obtained forT = 30 d and very close to the exact values.

Fitting to observed root distributions

MaizeThe fitted parameters are given in Table 3. Placementof fertiliser in the row resulted in a small and down-wards oriented rooting pattern, while for the case ofbroadcast fertiliser application a more circular rootingpattern was observed (Figure 5). For row application,the ratio of DZ over DX (the dimensionless parameterp in De Willigen et al., 2002) is 2.2, while for broad-cast application this ratio was 0.8. The correspondencebetween observed and simulated rooting patterns isgood (Figures 5 and 6).

In Figure 6, the correspondence between observedand fitted rooting patterns are shown for all cases.Since in practice the smallest value of C that can bemeasured on pinboard samples is about 0.1 cm cm−3

only the correspondence is shown for measured C>

0.1 cm cm−3.

Eastern white cedar and gladiolusFor eastern white cedar and gladiolus, the observedrooting patterns could effectively be simulated withthe numerical model (Figure 6), with the fitted para-meters as given in Table 3. Note that for these differentcrops quite different parameters were obtained. Forexample, the ratio of DZ over DR for eastern whitecedar equals 7.0, while for gladiolus the ratio of DZ

over DX is 0.5.

Tomato grown on rockwoolThe two data sets for tomato grown on rockwool couldbe fitted rather well (Figures 6 and 7). The fitted para-meters are different for the two cases. This is due to thetwo different cultivars involved, as well as the fact thatplanting configuration differed: for Sonata half the in-row plant distance was L = 25 cm while for Aromatathis was L = 12 cm (Table 2).

Root growth influenced by pressure head

The reduction function fh can have a pronounced effectas is seen from Figure 8. One should note that theseuniform extreme conditions in the soil persisted for100 days. Roots hardly penetrate in the right part of thesoil. This results in accumulation of roots below theplanting position (X,Z) = (0,0). The net effect of dif-ferent C gradients in the Z direction and the increasedroot decay in the left part of the soil column is thatroots penetrate slightly deeper in the left part of thesoil column than in case of uniform h.

We recognise that this is an extreme situation. Amore realistic situation can be found in dry climatic re-gions where water is supplied near the plants throughdrip irrigation. Dry soil zones will develop betweenplants, while underneath the plant wet regions willpersist.

Conclusions

In simulation models for water movement and nutrienttransport, uptake of water and nutrients by roots formsan essential part. As roots are spatially distributed,prediction of root growth and root distribution is cru-cial for modelling water and nutrient uptake. Althoughanalytical models describing rooting patterns are easyto use and give good insight in rooting patterns underspecial conditions, as was shown in the preceding pa-per by De Willigen et al. (2002), a numerical modelfor describing rooting patterns by using a diffusiontype partial differential equation is more flexible, es-pecially in situations where root input occurs insidethe rooting medium (which can not be handled by theanalytical model of De Willigen et al., 2002). The dif-fusion type equation can be used to mimic observedrooting patterns as was shown by a few examples withquite different crops and rooting media (Figure 6). Formaize, we used data for two different types of fertiliserapplication: broadcast and row application. In the caseof row application, roots extend more vertically thathorizontally with respect to the broadcast applicationsituation. This is reflected in a larger ratio of diffusioncoefficients DZ/DX.

The diffusion model to describe root growth, orbetter root distribution has no fundamental physiolo-gical background. Two biologically sound parametersare included: total biomass (length) production andspecific root decay rate, the latter being of the orderof say 0.01–0.1 d−1. The diffusion coefficients are

261

Figure 5. Fitted (top row) and measured (bottom row) root length density distributions C (cm cm−3) for maize for treatments row fertilisation(left column) or broadcast fertilisation (right column). The arrows indicate plant position.

Figure 6. Comparison between observed and fitted root length density distributions C (cm cm−3) for all examples. Only data with C >

0.1 cm cm−3 are shown, as this is practically the threshold value which still can be determined on pinboard analyses.

262

Figure 7. Fitted (left) and measured (right) root length density distributions C (cm cm−3) for tomato grown on rockwool. The arrows indicateplant position.

Figure 8. Comparison between simulated root distributions (left) in a case of uniform pressure head hA and (right) in case of non uniformpressure head with strong reduction of root penetration in the region with pressure head hB . The arrows indicate plant position.

263

empirical parameters. Their ratio indicates whetherthe rooting pattern remains shallow but wide, deepbut small, or develops similarly in all directions. Onecould consider these parameters as plant specific para-meters that are valid under normal (non-limiting) soilconditions. To mimic the plant’s reaction to limitingsoil conditions can then be done simply by introducingone or more reduction functions as demonstrated inthis paper.

In this paper, we considered constant root growthQ. It is relatively simple to adapt the code so that Q isa function of time and input to the model (Q being a‘fixed’ function of time). Somewhat more complicatedis the situation when a predefined function Q(T) is im-plemented in the code yielding additional parametersto be optimised. For example, Q(T) can be some lo-gistic (three additional parameters) or expo-linear (twoadditional parameters) function. The drawback of thelatter approach could be an over-parameterised model,resulting in difficulties in parameter optimisation.

Finally, it is generally known that non-linear para-meter optimisation does not always result in uniqueoptimised solutions. The solution will be dependenton the initial guesses of the parameters. Therefore,it is advised to do the optimisation for different setsof initial guesses. Moreover, our experience so far isthat the parameters Q and � seem to be highly cor-related. Thus, it is advisable to keep one of these twoparameters fixed at a constant value.

Acknowledgements

We are grateful to Phillip Ehlert, Gerard Brouwer,and Henk Pasterkamp for measuring and supplying theroot length density data for gladiolus, and to AnnettePronk for supplying the root length density data foreastern white cedar. Part of this research was fundedby INRA Department of Environment and Agronomy.The root length density data for tomato (Lycopersiconesculentum cv. Aromata) grown in rockwool were ob-tained within the Dutch subsidies program EET (Eco-nomy, Ecology and Technology) project EETK99038‘Hydrion-Line-II. On-line monitoring and control ofclosed greenhouse systems’.

Appendix A. Integration of Equations (1) and (2)

The control volume finite element method (Heinen andDe Willigen, 1998, 2001; Patankar, 1980) integrates

Equation (1) for each control volume. The details canbe found in above mentioned references. Here a shortdescription is presented. Integration of Equation (1),according to Equation (3), can be done term by term.For convenience C refers to the new time T+�T unlessdenoted with superscript T when it refers to the oldtime T. In what follows, the four integrals should beinterpreted as:∫

X

∫Y

∫Z

∫T

=∫ Xi,j

Xi−1,j

∫ Y+�Y

Y

∫ Zu,j

Zi,j−1

∫ T +�T

T

cartesian co-ordinate system∫R

∫ϕ

∫Z

∫T

=∫ Ri,j

Ri−1,j

∫ 2π

0

∫ Zu,j

Zi,j−1

∫ T +�T

T

cylindrical co-ordinate system. The first term yields:∫X

∫Y

∫Z

∫T

−∂C

∂TdT dZ dY dX

= −(CI,J − CTI,J)�XI�Y�ZJ (A-1)

For the diffusive flux in the X-direction we have:∫X

∫Y

∫Z

∫T

∂

∂X

(DX

∂C

∂X

)dT dZ dY dX

=[DX

∂C

∂x

]Xi,j

Xi−1,j

�Y �ZJ �T (A-2)

The term between the square brackets on the righthand side of Equation (A-2) represents the flux acrossa vertical boundary of a CV. The gradient in C acrossthis boundary is, temporarily, assumed to changelinearly with distance. Then Equation (A-2) becomes:∫

X

∫Y

∫Z

∫T

∂

∂X

(DX

∂C

∂X

)dT dZ dY dX

=[

DXi,j

CI+1,J − CI,J

δXi− DXi−1,J

CI,J − CI−1,J

δXi−1

]× �Y �ZJ �T (A-3)

When DX differs from CV to CV, e.g. due to differ-ent environmental conditions as stated by for instanceEquation (20) in the text, the diffusion coefficientat the CV interface is obtained by distance-weightedgeometrical averaging, according to:

DXi,j = DfX,iX,I,JD

1−fX,iX,I+1,J (A-4)

where fX,i is the distance weighing factor for the X-direction defined by:

fX,i = �XI+1

�XI − �XI+11(A-5)

264

By analogy, for the diffusive flux in the Z-direction wehave:∫

X

∫Y

∫Z

∫T

∂

∂Z

(DZ

∂C

∂Z

)dT dZ dY dX

=[

DZ∂C

∂Z

]Zi,j

Zi,j−1

�XI�Y�T

=[

DZi,j

CI,J+1 − CI,J

δZj− DZi,j−1

CI,J − CI,J−1

δZj−1

]× �XI �Y �T (A-6)

The diffusion coefficient at a horizontal interface isgiven by:

DZi,j = DfZ,jZ,I,JD

1−fZ,j

Z,I,J+1, (A-7)

where fZ,j is the distance weighing factor for the Z-direction defined by:

fZ,j = �ZJ+1

�ZJ + �ZJ+1(A-8)

The decay term yields:∫X

∫Y

∫Z

∫T

−�C dT dZ dY dX

= −�CTI,J�XI�Y�ZJ�T (A-9)

Finally, the source term yields:∫X

∫Y

∫Z

∫T

Q dT dZ dY dX = QI,J�XI�Y�ZJ�T

(A-10)

Adding the terms on the right hand sides of Equa-tions (A-1), (A-3), (A-6), (A-9) and (A-10), usingsome simple algebra, and introducing the A coeffi-cients given by Equations (6) and (7) in the text, onefinally obtains Equation (5) as given in the text.

For the cylindrical co-ordinate system the proced-ure is analogous, except that care needs to be taken ofthe presence of the R co-ordinate in all terms. Withoutgiving further details, the respective terms yield:∫

R

∫ϕ

∫Z

∫T

− R∂C

∂TdT dZ dϕ dR

= −(CI,J − CTI,J )

(R2i − R2

i−1)

22π�ZJ

(A-11)

∫R

∫ϕ

∫Z

∫T

∂

∂R

(RDR

∂C

∂R

)dT dZ dϕ dR

=[

RiDRi,j

CI+1,J − CI,J

δRi− Ri−1DRi−1,j

CI,J − CI,1−J

δRi−1

]× 2π�ZJ�T (A-12)

∫R

∫ϕ

∫Z

∫T

R∂

∂Z

(DZ

∂C

∂Z

)dT dZ dϕ dR

=[

DZi,j

CI,J+1 − CI,J

δZj− DZi,j−1

CI,J − CI,J−1

δZj−1

]

× (R2i − R2

i−1)

22π�T (A-13)

∫R

∫ϕ

∫Z

∫T

− R�C dT dZ dϕ dR

= −�CTI,J

(R2i − R2

i−1)

22π�ZJ�T (A-14)

and:∫R

∫ϕ

∫Z

∫T

RQ dT dZ dϕ dR

= QI,J(R2

i − R2i−1)

22π�ZJ�T (A-15)

Appendix B. Analytical expression for total rootlength

De Willigen et al. (2002) gave an analytical expres-sion for the root distribution. Here it was presentedin dimensional form: Equations (16) through (19) inthe text. The total root length Ctot (cm) in the region0<X<L and 0<Z<ZL can be obtained by integratingEquation (16):

Ctot =∫ Y+�Y

Y

∫ L

X=0

∫ ZL

Z=0

(C0

π+ 2

π

∞∑n=1

Cn cos[αnX])

dZ dX dY = �YL

π

∫ ZL

z=0C0 dZ (B-1)

The cos-term in the integral results in a sin-term tobe evaluated at 0 and nπ (with n any integer), whichis zero. Thus the second term of Equation (16) dis-appears and only the C0-term remains, which is a

265

function of Z only (Equation (B-1)). Ctot can becomputed from:

Ctot = L�YQL

2�

2(

1 − exp(−�T)erf(

ZL2√

DZT

))+

−exp(−

√�DZ

ZL

) (1 − erf

(ZL

2√

DZT− √

�T))

+−exp

(√�DZ

ZL

) (1 − erf

(ZL

2√

DZT+ √

�T))

(B-2)

For ZL → ∞, Equation (B-2) reduces to:

Ctot = L�YQL

�(1 − e−�T) (B-7)

which also follows from integrating Equation (1) of DeWilligen et al. (2002) including corresponding bound-ary conditions over the complete domain 0<X<L and0<Z<∞.

References

Acock B and Pachepsky Ya A 1996 Convective-diffusive model oftwo-dimensional root growth and proliferation. Plant Soil 180,231–240.

De Willigen P and Van Noordwijk M 1987 Roots, Plant Pro-duction and Nutrient Use Efficiency. PhD Thesis, WageningenUniversity, Wageningen, The Netherlands. 282 pp.

De Willigen P, Heinen M, Mollier A and Van Noordwijk M 2002Two-dimensional growth of a root system modelled as a diffusionprocess. I. Analytical solutions. Plant Soil 240, 225–234.

Do Rosário M, Oliveira G, Van Noordwijk M, Gaze S R, Brouwer G,Bona S, Mosca G and Hairah K 2000 Auger sampling, ingrowthcores and pinboard methods In Root Methods. Eds. Smit A L,Bengough A G, Engels C, Van Noordwijk M, Pellerin S and VanDe Geijn S C. pp. 175–210. Springer, Berlin.

Hayhoe H 1980 Analysis of a diffusion model for plant root growthand an application to plant soil-water uptake. Soil Sci. 131, 334–343.

Heinen M 2001 FUSSIM2: Brief description of the simulationmodel and application to fertigation scenarios. Agronomie 21,285–296.

Heinen M and De Willigen P 1998 FUSSIM2. A Two-DimensionalSimulation Model for Water Flow, Solute Transport and RootUptake of Water and Nutrients in Partly Unsaturated Porous Me-dia, Quantitative Approaches in Systems Analysis No. 20, DLOResearch Institute for Agrobiology and Soil Fertility and the C.T.de Wit Graduate School for Production Ecology, Wageningen,The Netherlands. 140 pp.

Heinen M and De Willigen P 2002 FUSSIM2 version 5. New Fea-tures and Updated user’s guide. Alterra rapport 363, Alterra,Wageningen. 164 pp.

Patankar S V 1980 Numerical Heat Transfer and Fluid Flow.Hemisphere Publishing Corporation, New York. 197 pp.

Press W H, Teukolsky S A, Vetterling W T and Flannery B P 1996Numerical Recipes in Fortran 77, the Art of Scientific Comput-ing (vol. 1 of Fortran Numerical Recipes). Cambridge UniversityPress, New York. 933 pp.

Pronk A A, De Willigen P, Heuvelink E and Challa H 2002 Devel-opment of fine and coarse roots of Thuja occidentalis ‘Brabant’in non-irrigated and drip irrigated field plots. Plant Soil 243,161–171.

Schröder J J, Groewold J and Zaharieva T 1996 Soil mineral nitro-gen availability to young maize plants as related to root lengthdensity distribution and fertilizer application method. Neth. J.Agric. Sci. 44, 209–225.

Schröder J J, The Holte L and Brouwer G 1997 Response of silagemaize to placement of cattle slurry. Neth. J. Agric. Sci. 45, 249–261.

Van Noordwijk M 1978 Zoutophoping en beworteling bij de teeltvan tomaten op steenwol (in Dutch; with a summary: Distribu-tion of Salts and Root Development in the Culture of tomatoeson Rockwool). Report 3–78, Institute for Soil Fertility Research,Haren, The Netherlands. 21 pp.

Visual Numerics Inc 1997 IMSL Math Library Volumes 1 and 2.9990 Richmond Avenue, Suite 400, Houston, Texas, 77042-4548, USA.

Wesseling J 1991 Meerjarige Simulatie van Grondwaterstromingvoor Verschillende Bodemprofielen, Grondwatertrappen en Ge-wassen met het Model SWATRE (in Dutch). Rapport 152,Staring Centrum, Wageningen. 63 pp.

Section editor: B.E. Clothier