A vertically extended stochastic model of soil moisture in the root zone

Comprehension of the soil moisture dynamics in the rhizosphere is fundamental in the study of the water balance, vegetation productivity in water-limited ecosystems, soil microbial activity, and atmospheric boundary layer dynamics. This paper develops a vertically extended model for soil moisture, which accounts for (1) the stochasticity of precipitation, (2) the vertical distribution of plant roots, and (3) the (linear or nonlinear) dependence of transpiration on soil water content. The stochastic model is mathematically formulated under a few simplifying hypotheses, and the depth-dependent steady state probability density function of soil moisture is obtained analytically. The dependence of these solutions on climate conditions is discussed through some exemplifying cases.