Galerkin Dynamical Modeling of Porous Medium Convection using an Eigenbasis from Upper Bound Theory

Galerkin projection is a common strategy for generating ODE models of PDE systems, either as a means of performing highly resolved direct numerical simulation (DNS) or as a method for generating reduced-order dynamical models. Popular bases for the associated spectral expansions include Fourier and Chebyshev modes, eigenfunctions of linear stability operators about ``laminar'' base solutions or empirical mean flows, or modes arising from Proper Orthogonal Decomposition (POD) of numerical or experimental system realizations. Here we employ an alternative, fully a-priori spectral basis composed of eigenfunctions from upper bound theory, for both fully resolved and reduced dynamical modeling of porous medium convection -- a system that is receiving increased attention owing to applications in CO2 sequestration in terrestrial aquifers. Because this new basis is naturally adapted to the dynamics at a given Ra, our DNS requires a fraction of the total number of modes used in traditional (e.g. Fourier) spectral Galerkin simulations. Moreover, for ``moderate'' Rayleigh numbers ($Ra \la 10^3$) we demonstrate that mode slaving can be used to further reduce the dimension of the truncated dynamical systems.