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 ( 2 k ) . 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 ( Pe 2 ) . 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.