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

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 [Formula: see text] decay quickly to 0 in less than 3 days while [Formula: see text] and Cd gradually decrease in more than 15 days; the depletion profile spread of [Formula: see text] is apparently narrower than [Formula: see text] 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 [Formula: see text] and Cd 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.