Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Abstract

Steady states of localized activity appear naturally in uniformly driven, dissipative systems as a result of subcritical instabilities. In the usual setting of an infinite domain, branches of such localized states bifurcate at the subcritical "pattern-forming" instability and intertwine in a manner often referred to as "homoclinic snaking." In this paper we consider an extension of this paradigm where, in addition to the pattern-forming instability (with nonzero wavenumber), a large-scale neutral mode exists, having zero growth rate at zero wavenumber. Such a situation naturally arises in the presence of a conservation law; we give examples of physical systems in which this arises, in particular, thermal convection in a horizontal fluid layer with a vertical magnetic field. We introduce a novel scaling that allows the derivation of a nonlocal Ginzburg-Landau equation to describe the formation of localized states. Our results show that the existence of the large-scale mode substantially enlarges the region of parameter space where localized states exist and are stable.

A mechanism for streamwise localisation of nonlinear waves in shear flows

“…This is reminiscent of the blending of separate regions of regular and semi-infinite snaking into a unique region of semi-infinite snaking observed for some parameter values of the Swift-Hohenberg equation (Burke & Knobloch 2006). Furthermore, the existence of a conserved quantity, which in our case is mass-flux, accounts for the slanted arrangement of the leftmost saddle-nodes (Dawes 2008;Beaume et al 2013a), thus allowing for localised solutions to exist at lower and higher values of the parameter than is possible in regular snaking. Physically, this means that while the multiplicity of spanwise localised solutions of different lengths is restricted to a finite Re-range (where the bends exist, within the broadended Maxwell point), a surging number of streamwise localised solutions of varying length extending to high Re appears as longer domains are considered.…”

“…Although snaking has originally been discovered and studied in variational systems, it is a well known fact that the problem needs not be variational in time to produce this type of behaviour, as demonstrated for dissipative systems such as plane Couette flow (Schneider et al 2010). Finally, the introduction of a large scale neutral mode (Dawes 2008), associated to the existence of a conservation law, results in a substantial enlargement of the region of existence of localised states and the snaking branches becomes slanted. Two-dimensional PPF conserves massflux (or, alternatively, mean pressure gradient), which justifies the slanted arrangement of the leftmost saddle-nodes even if snaking has been disrupted and isolated branches of localised solutions have emerged.…”

Localized rotating convection with no-slip boundary conditions

“…It is also known that the form of the resulting bifurcation diagram is significantly altered in the presence of a conserved quantity. 15,16 This is because the localized structures typically expel the conserved quantity thereby raising its magnitude in the region outside. This in turn modifies the background state and leads to so-called slanted snaking.…”

“…This in turn modifies the background state and leads to so-called slanted snaking. [15][16][17][18] The presence of slanted snaking implies that localized states are present over a much wider interval in parameter space than is the case with standard snaking. Slanted snaking is a consequence of a conserved quantity, such as imposed magnetic flux in magnetoconvection 19 or fixed zonal momentum in rotating convection with stress-free boundary conditions at top and bottom, 18 and is a finite size effect -in an unbounded domain the conserved quantity exerts no effect and the system reverts to standard snaking.…”

Force-chain buckling in granular media: a structural mechanics perspective

“…resembles the localized shape of figure 7, while the other has three separate humps. Each develops a form of the snaking response (Burke & Knobloch 2007;Dawes 2008) seen in a number of general situations when cells buckle locally and restabilize in turn (Hunt et al 2000). This sequence is illustrated in more detail for solution ➁ in figure 9.…”