Approximate analytical solutions of root surface nutrient uptake flux and rhizosphere solute concentrations

Aims We use the Nye-Tinker-Barber model to obtain approximate analytical solutions of the root surface nutrient uptake flux and the rhizosphere solute concentrations.
Methods The rhizosphere is divided into the distant field and the close-range field. The outer solution is obtained by the similarity variable method, and the inner solution is obtained by the rescaling method. Expanding the outer solution from root surface and matching it with the inner solution, we finally obtain the approximate analytical solutions of the root surface nutrient uptake flux and rhizosphere solute concentrations based on Nye-Tinker-Barber model. Numerical simulations are performed on the uptake fluxes of six elements (N, K, P, Mg, S and Ca) and the solute concentrations of two elements (N and K). Comparisons are made among the Nye- Tinker-Barber model approximate analytical solutions, Roose’s analytical solutions and the numerical solutions.
Important findings The approximate analytical solutions of the root surface uptake fluxes based on the diffusion Nye-Tinker-Barber model are similar to those of the Roose's analytical solutions; both approaches produced higher values than the numerical solutions. The approximate analytical solutions of the rhizosphere N and K concentrations based on the diffusion Nye-Tinker-Barber model are similar to the Roose’s analytical solutions and coincide with the trend of changes in the numerical solutions. The approximate analytical solutions of all elements but N, of the root surface, uptake fluxes based on the convection-diffusion Nye-Tinker-Barber model are more similar to the numerical solutions than the Roose’s analytical solutions, and the approximate analytical solutions of the rhizosphere N and K concentrations are coincide with the trend of changes in the numerical solutions.