Approximate analytical solution for soil chemical transfer to runoff: A modified boundary condition

Two mass balance equations were used to model the transfer of dissolved chemicals from the soil solution to the surface runoff water and the transport of these chemicals to the field outlet. One mass balance equation was written for chemicals dissolved in the overland water, the other for chemicals within the soil profile. Chemical input into the surface water (upper boundary condition) was expressed as a rate‐limited convective mass transfer, depending on both soil surface and runoff concentrations. Isolating a slow and fast time scale and scaling the mass balance equations to the slow one yielded a parameter, ε, which multiplies the time derivative of the mass balance equation written for overland flow. In most cases ε ≪ 1, providing a singular perturbation problem that was solved by using the method of matched asymptotic expansion. The approximate solution, uniformly valid over the entire domain, was made up of two terms: a leading‐order solution and a first‐order solution, the latter of which was relatively small, even for ε = O(1). The leading‐order solution was compared with that for a simpler case, in which the convective mass transfer (upper boundary condition) depends only on the soil surface concentration. The comparison indicated those limited cases to which the simpler boundary condition can be applied resulting in a very small error. Although it is not possible to get a strictly analytical solution for a problem involving a modified upper boundary condition, the approximate analytical solution is easily obtained.