On the nonlinear dynamics of a saline boundary layer formed by throughflow near the surface of a porous medium

We consider gravitational instability of saline boundary layers, observed at the subsurface of salt lakes. This boundary layer is the result of the convective transport induced by the evaporation at the horizontal surface of a confined porous medium. When this upward transport is balanced by salt dispersion, a steady state boundary layer is formed. However, this boundary layer can be unstable when perturbed. This results in complex groundwater motion and density fields. The aim of this paper is to investigate the existence of finite amplitude solutions describing these resulting patterns (both the number of solutions and their structure), their stability, and their dependency on the system Rayleigh and Péclet numbers. For this purpose we construct a low-dimensional dynamical system (a reduced model) by projecting the nonlinear model equations onto a relatively small set of eigenfunctions of the problem linearized at criticality. The Galerkin projection approach is complicated by the fact that the problem under consideration is non-self-adjoint due to the existing evaporation. This implies that the eigenfunctions do not form an orthogonal set and therefore the adjoint eigenfunctions are used for the projection. The reduced model is constructed in such a way that it is capable of providing solutions in the strongly nonlinear regime as well. Convergence of these solutions towards the fully nonlinear model results is shown by means of direct numerical simulations. Further, the reduced model seems to partly capture the complex nonlinear behavior as seen in Hele–Shaw experiments by Wooding et al. [R.A. Wooding, S.W. Tyler, I. White, P.A. Anderson, Convection in groundwater below an evaporating salt lake: 2. evolution of fingers or plumes, Water Resour. Res. 33 (6) (1997) 1219–1228]. The physical transition mechanism that explains the occurrence of some observed bifurcation types is presented as well.