Localization and advectional spreading of convective currents under parametric disorder

TL;DR

This paper investigates how parametric disorder causes localized convective patterns in a porous layer and how advection influences their delocalization, with implications for pattern formation in dissipative systems.

Contribution

It introduces a study of localization and advection effects on convective patterns in a disordered porous medium, connecting to Anderson localization and nonlinear dynamics.

Findings

Weak advection causes upstream delocalization of flow patterns.

Transition from localized to global flow occurs near the instability threshold.

Results are applicable to a broad class of dissipative media.

Abstract

We address a problem which is mathematically reminiscent of the one of Anderson localization, although it is related to a strongly dissipative dynamics. Specifically, we study thermal convection in a horizontal porous layer heated from below in the presence of a parametric disorder; physical parameters of the layer are time-independent and randomly inhomogeneous in one of the horizontal directions. Under such a frozen parametric disorder, spatially localized flow patterns appear. We focus our study on their localization properties and the effect of an imposed advection along the layer on these properties. Our interpretation of the results of the linear theory is underpinned by numerical simulation for the nonlinear problem. Weak advection is found to lead to an upstream delocalization of localized current patterns. Due to this delocalization, the transition from a set of localized patterns to an almost everywhere intense "global" flow can be observed under conditions where the disorder-free system would be not far below the instability threshold. The results presented are derived for a physical system which is mathematically described by a modified Kuramoto-Sivashinsky equation and therefore they are expected to be relevant for a broad variety of dissipative media where pattern selection occurs.