The coexistence of multiple stationary, spatially localized structures has recently been reported for convection in an inclined porous layer, but the influence of imperfectly conducting boundaries has not been studied. This paper analyzes the traveling behavior of asymmetric, spatially localized convective structures (“convectons”), consisting of one or more pulses, in a two-dimensional inclined layer of a porous medium subject to a fixed temperature at the bottom and an imperfectly conducting boundary at the top, such that midplane reflection symmetry is broken. Extensive direct numerical simulations (DNS) are conducted for a wide range of Biot numbers associated with the upper boundary, revealing nontrivial relationships between the drift velocity $c$ of spatially localized structures and the symmetry-breaking parameter $\kappa \ge 0$ based on the Biot number, with $\kappa =0$ corresponding to perfect midplane reflection symmetry associated with $c=0$. In small domains, the drift velocity $c$ is positive (corresponding to upslope motion) and increases monotonically with $\kappa$, while for localized structures in larger domains consisting of a small number of pulses $c$ can be of either sign depending on parameters. For longer structures $c$ reverts to positive values and again increases monotonically with $\kappa$. We show that the along-slope tails of pulses and the associated long-range interactions are governed by the dominant spatial eigenvalues, whose real part is of the smallest magnitude, and we uncover a transition at a finite symmetry-breaking strength $\kappa ={\kappa }_c>0$: for $\kappa <{\kappa }_c$, both dominant eigenvalues are complex, implying that the upslope and downslope exponential tails are oscillatory. In contrast, for $\kappa >{\kappa }_c$, the dominant spatial eigenvalue with a positive real part becomes real, implying that the downslope exponential tail transitions from an oscillatory profile to a monotonic one. As a result, for $\kappa <{\kappa }_c$, bound states consisting of different numbers of pulses are present. These display rich dynamical phenomena including inelastic collisions leading to other bound states, while for $\kappa \gg {\kappa }_c$, adjacent pulses are found to repel one another and so tend to spread out, eventually becoming equispaced in the finite computational domain. The strength of the repulsive interaction increases with $\kappa$, i.e., with increased symmetry breaking. A reduced description of this behavior is proposed based on the interaction between tails of adjacent localized structures, which accurately reproduces the repulsion and inelastic collisions observed in DNS. The reduced model indicates that the transition from bound states to an equidistant configuration occurs when the monotonic tail and the oscillatory tail have the same slope, which occurs at a value of $\kappa$ greater than ${\kappa }_c$. The results presented here represent a step towards understanding the dynamics of spatially localized patterns in moderate-Rayleigh number convection in an inclined porous medium subject to an imperfectly conducting boundary.

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• Convection in porous media
• Pattern formation
• Traveling waves