Plant and Soil 240: 225–234, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 225 Two-dimensional growth of a root system modelled as a diffusion process. I. Analytical solutions P. de Willigen 1,4 , M. Heinen 1 , A. Mollier 2 & M. Van Noordwijk 3 1 Alterra, Dept. of Water and the Environment, P.O. Box 47, 6700 AA Wageningen, the Netherlands; 2 INRA, Unit´ e d’Agronomie, 71, Avenue Edouard-Bourlaux, B.P. 81, 33883 Villenave d’Ornon Cedex, France; 3 International Centre for Research in Agroforestry (ICRAF), Bogor, Indonesia, P.O.box 161, Bogor, 16001, Indonesia; 4 Corresponding author ∗ Received 6 April 2001. Accepted in revised form 13 January 2002 Key words: mathematical solutions, modeling, root length density Abstract For functioning of a root system, the temporal development of distribution of roots in the soil is important. For example, for computing uptake of water and nutrients the root length density distribution might be required. A way to describe root proliferation is to consider it as a diffusion process with a first-order sink term accounting for decay. In this paper, analytical solutions are derived for two-dimensional diffusion of roots both in a rectangular area, and in a cylindrical volume. The source of root dry matter is located at the surface. Root dry matter enters the soil domain through a part of the soil surface. It is shown that different distribution patterns are obtained, with different ratios of the diffusion coefficients in horizontal and vertical direction. From the solutions obtained it can be shown that for the situation where the dry matter enters through the complete surface eventually a steady-state occurs where root length density decreases exponentially with depth, as often is found in experiments. Introduction To judge the uptake potential of a root system, not only its size (i.e., total mass or total length) but also its distribution pattern should be known. Likewise in simulation models on crop growth, it is not only necessary to know which fraction of the dry matter increase is allocated to root mass increase, but also how this increase is distributed over the rootable zone. One way to achieve this is to describe root growth and distribution as a result of a diffusion type process. The first who did this were Gerwitz and Page (1974). Later, Hayhoe (1981) presented a more detailed pa- per, whereas Acock and Pachevsky (1996) extended the theory to two dimensions and compared theoretical (obtained by numerical methods) with experimentally determined distributions. It is the aim of this paper to present analytical solutions to the root diffusion equation in two dimen- ∗ FAX No: +31(0)317-419000. E-mail: p.dewilligen@alterra.wag_ur.nl sions and two geometries, a rectangular area and a cylindrical volume. In subsequent papers, numerical extensions of the theory will be discussed, and the ap- plication of theory to experimental data will be shown. In making the derivations, it has been assumed that root growth is not limited by drought or flooding, that the soil is homogeneous, that aeration is sufficient and that root growth takes place at constant temperature. Mathematical background Rectangular geometry Consider a crop sown in a row perpendicular to the X-direction, the row has a width of 2X 1 cm, the dis- tance between the rows is 2L cm (Figure 1). In the Z-direction (downward), the region is considered to be infinite. If the extension of the root system can be described as diffusion, the next partial differential