Approximate nutrient flux and concentration solutions of the Nye–Tinker–Barber model by the perturbation expansion method

Abstract

The Nye–Tinker–Barber model is a basic and representative one for single-ion nutrient uptake by plant root from the soil and we aim to derive its approximate analytical solutions of flux and concentration. We divide the rhizosphere into the inner and the outer fields, match the inner and the outer solutions near the root surface, and then obtain the approximate analytical solutions of nutrient uptake flux at the root surface and global nutrient concentration of the diffusion or the convection-diffusion Nye–Tinker–Barber model. The analytical and numerical fluxes of K+ and H2PO4− decay quickly to 0 in less than 3 days while NO3− and Cd2+ gradually decrease in more than 15 days; the depletion profile spread of H2PO4− is apparently narrower than NO3− and K+ in 24 days. The different flux and concentration patterns of 4 nutrients result from their mobility and solubility in the rhizosphere. In comparison with the numerical simulations and the previous analytical results, we find that the analytical flux will overestimate the numerical flux of NO3− and Cd2+ while the analytical concentration can accurately predict the numerical concentration; the flux and the concentration solutions of the convection-diffusion Nye–Tinker–Barber model can be simplified to the diffusion versions by the Péclet number, and they can more widely describe the transport of nutrients of different attributes in soils of different textures with different levels of saturation, conductivity and permeability.