Insights from a pseudospectral approach to the Elder problem

Abstract

The aim of this paper is to clarify and circumvent the issue of multiple steady state solutions in the Elder problem. A pseudospectral method is used to avoid numerical error associated with spatial discretization. The pseudospectral method is verified by comparison to an analytical solution at Rayleigh number, Ra = 0, and by reproducing the three stable steady state solutions that are known to exist at Ra = 400. A bifurcation diagram for 0 < Ra < 400, which is free of discretization error, confirms that multiple steady states are indeed an intrinsic characteristic of the Elder problem. The existence of multiple steady states makes the Ra = 400 Elder problem less suitable for benchmarking numerical models. To avoid the multiple steady states, we propose a benchmark at Ra = 60. The results for this Low Rayleigh Number Elder Problem are presented and compared to simulations with the commercial groundwater modeling package FEFLOW. Correspondence between the pseudospectral model and FEFLOW is excellent.