Convectons in a rotating fluid layer

Abstract

Two-dimensional convection in a plane layer bounded by stress-free perfectly conducting horizontal boundaries and rotating uniformly about the vertical is considered. Time-independent spatially localized structures, called convectons, of even and odd parity are computed. The convectons are embedded within a self-generated shear layer with a compensating shear flow outside the structure. These states are organized within a bifurcation structure called slanted snaking and may be present even when periodic convection sets in supercritically. These interesting properties are traced to the presence of a conserved quantity and hence to the use of stress-free boundary conditions.

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.…”

Localized rotating convection with no-slip boundary conditions

“…In previous studies, 18,30,31 stress-free, fixed temperature boundary conditions have been applied at the upper and lower boundaries, i.e.,…”

“…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.…”

“…[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. 17 In this paper, we seek to understand the transformation from slanted to standard snaking as the conservation of the conserved quantity is progressively broken and the associated changes in the width of the parameter interval within which convectons may be found.…”

“…17 In this paper, we seek to understand the transformation from slanted to standard snaking as the conservation of the conserved quantity is progressively broken and the associated changes in the width of the parameter interval within which convectons may be found. For this purpose, we select convection in a rotating layer where slanted snaking has recently been studied 18 and employ homotopic continuation of the boundary conditions from stress-free boundary conditions for which slanted snaking is present to no-slip boundary conditions. The procedure allows us to (a) show that convectons in a rotating layer are present even in the presence of the more realistic no-slip boundary conditions, (b) determine the width of the Rayleigh number interval within which they are found, (c) track the morphology changes in the associated bifurcation diagram as the boundary conditions are deformed, and (d) quantify the influence of the conserved zonal momentum on the behavior of this system.…”