Magnetohydrodynamic convectons

Abstract

Numerical continuation is used to compute branches of spatially localized structures in convection in an imposed vertical magnetic field. In periodic domains with finite spatial period, these branches exhibit slanted snaking and consist of localized states of even and odd parity. The properties of these states are analysed and related to existing asymptotic approaches valid either at small amplitude (Cox and Matthews, Physica D, vol. 149, 2001, p. 210), or in the limit of small magnetic diffusivity (Dawes, J. Fluid Mech., vol. 570, 2007, p. 385). The transition to standard snaking with increasing domain size is explored.

Localized rotating convection with no-slip boundary conditions

“…In the presence of stress-free boundary conditions, the system admits a conservation law and it proved useful to compare solutions using averaged quantities such as E. This approach proved effective at capturing the slope of the slanted branches and helped establish the conclusion that in the presence of a conserved quantity the convecting region at fixed Rayleigh number always fills the same fraction of the available domain, regardless of the domain size, provided only that it is finite. 17,18 For example, at a specific value of the Rayleigh number, 3-roll localized solutions may form in one domain, while 6-roll convectons would form in a domain twice as large (see Figs. 4 and 5 in Ref.…”

“…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.…”

“…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. 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.…”

Spatial localization in heterogeneous systems

“…In this representation the localized states are almost exactly vertically aligned, as in standard snaking, and the snaking interval is almost independent of α, at least for small |α|, indicating that structures such as that shown in Fig. 24 resembling slanted snaking are indeed the result of a slow variation of the effective forcing parameter, much as in systems with a conserved quantity when these are defined on a periodic domain with a finite period [6,33]. For larger values of |α| the vertical alignment is less good, an observation we attribute to the fact that as N and hence r increases the front profile also changes; as a result the front location x c , as constructed above, does in fact depend weakly on r, and this dependence is expected to grow with increasing |α|.…”

“…These include different convective systems driven by an imposed temperature difference [1][2][3][4][5][6][7], a ferrofluid subject to an imposed magnetic field [8] and an optical light valve experiment driven by a nominally uniform light intensity [9]. Other systems exhibiting localized structures include shear flows [10,11], gas discharges [12], and a variety of optical configurations [13,14].…”