Damping of sinusoidal surface flux fluctuations with soil depth

Sinusoidal fluctuations of the flux at the soil surface dampen with depth in the vadose zone such that beyond a certain depth, flow may be approximated as steady. To investigate the damping with depth, we developed a new analytic solution for vertical periodic flow in the vadose zone, with a sinusoidal flux specified at the soil surface. The solution is based on the use of the Gardner-Kozeny model for hydraulic conductivity and soil moisture and on a linearization of the diffusive and advective terms in the governing differential equation. A characteristic length is presented for the damping of the flux with depth. At a depth of three times the characteristic length, the amplitude of the flux had reduced to 5% of the value at the surface. We compared the analytic solution to finite-element solutions of the original, nonlinear differential equation for 16 soils based on four reference soils, using a daily sinusoidal cycling of evaporation and infiltration at the soil surface. For the soils and circumstances investigated, the analytic solution produced reasonable values of the damping factor at any depth in the soil profile compared with the finite-element solutions. The solution is more accurate when the fluctuations (both amplitude and period) are smaller. The presented solution may be used for general cases of one-dimensional infiltration when the surface flux is writien as a Fourier series.