Stability analysis of the unsaturated water flow equation 1. Mathematical derivation

This paper provides a theoretical stability analysis of gradual wetting fro nts based on perturbation analysis. A traveling wave solution of the one-di mensional vertical flow Richards' equation is used as the basic flow on whi ch three-dimensional perturbations are introduced. By locally linearizing t he diffusivity form of the three-dimensional Richards' equation a linear pa rtial differential equation is obtained which governs the perturbation vari ables. The stability of each point at the wetting front is considered in a local coordinate system. The analysis of this perturbation equation at thes e points of the wetting front provides not only the relationship between th e finger sizes and the nonponding infiltration rates at the soil surface bu t also the traveling speeds of the fingers rooted from these points. Once a perturbation is introduced at some point on the wetting front, there are t hree possibilities for the development of the perturbation. (1) The perturb ation will monotonically decline with time; in this case, no fingers will f orm and the system is stable. (2) The perturbation does ndt decline with ti me, but its downward velocity is less than that of the stable basic wetting front; thus the distribution layer will gradually cover the fingers and th e system will become stable. (3) The perturbation will increase with time a nd have a downward velocity greater than that of the stable wetting front; in this case, the finger will persistently grow in front of the stable wett ing front and the system will become unstable. This analysis can be applied to an unsaturated homogeneous soil profile with uniform initial water cont ent for the prediction of instability and for the estimation of finger char acteristics over a wide range of infiltration rates.