Existence of orthogonal domain walls in B{\'e}nard-Rayleigh convection

TL;DR

This paper proves the existence of orthogonal domain walls in Bénard-Rayleigh convection, showing heteroclinic connections between orthogonal convective roll patterns through analytical and variational methods.

Contribution

It demonstrates the existence of a family of heteroclinic connections between orthogonal roll patterns in convection, extending previous reduced system analyses.

Findings

Existence of heteroclinic orbits connecting orthogonal convective rolls.

Analytical proof of heteroclinic connections under symmetry constraints.

Wave numbers linked to roll shifts parallel to the wall.

Abstract

In B{\'e}nard-Rayleigh convection we consider the pattern defect in orthogonal domain walls connecting a set of convective rolls with another set of rolls orthogonal to the first set. This is understood as an heteroclinic orbit of a reversible system where the x-coordinate plays the role of time. This appears as a perturbation of the heteroclinic orbit proved to exist in a reduced 6-dimensional system studied by a variational method in [3], and studied analytically in [10]. We then prove for a given amplitude, and an imposed symmetry in coordinate y, the existence of a one parameter family of heteroclinic connections between orthogonal sets of rolls, with wave numbers (different in general) which are linked to an adapted ''shift'' of rolls parallel to the wall.