Eckhaus instability in the Sutton problem: Dynamics of two interacting modes

This paper is motivated by the desire of solving the isolated case where the weakly nonlinear analysis proposed by Capone et al., [“A weakly nonlinear analysis of the effect of vertical throughflow on Darcy-Bénard convection,” Phys. Fluids 35, 014107 (2023)] turns out to be inadequate. As the authors point out in their paper, a resonance phenomenon occurs because the forcing term at second order has a component along a fundamental mode of the linear operator. This is consequence of the fact that those wavenumbers where the analysis fails are such that Ra(k)=Ra(2k). Therefore, two modes share the same critical Rayleigh number and are then generated and interact at the onset. In this paper, a weakly nonlinear analysis of a weak mass flow across a two-dimensional box of porous medium is presented in order to investigate the local bifurcation dynamics of two competing modes at the onset. Our analysis relies on the key assumption of a weak throughflow modeled by Pe≪1, allowing the Péclet number to be employed as small expansion parameter. The weakly unstable regime is then achieved by perturbing the critical Rayleigh number by O(Pe2). This choice enables the analytical determination of a pair of coupled amplitude equations, whose steady solutions are then investigated numerically using a continuation method. In a first instance, the analysis is developed at a specific wavenumber, while afterwards, modulation over a slow lengthscale is introduced and the study of Eckhaus instability is presented by tuning the deviation parameter κ characterizing the modulation. The resulting bifurcation diagrams revealing a rich dynamics with up to seven equilibria are analyzed and discussed.