Two-dimensional growth of a root system modelled as a diffusion process. I. Analytical solutions

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.