Mode Interactions in Large Aspect Ratio Convection

Hirschberg, P.; Knobloch, Edgar

doi:10.1007/s003329900039

Cited by 28 publications

(42 citation statements)

References 14 publications

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“…5(a). As soon as β = 0 this picture changes dramatically, as described in [19] and references therein: the crossings between the even neutral curves breakup, as do the crossings between odd neutral curves. Only crossings between opposite parity neutral curves are structurally stable.…”

Section: Linear Theorymentioning

confidence: 88%

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Homoclinic snaking in bounded domains

Houghton

Knobloch

2009

Phys. Rev. E

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Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of "snaking without bistability", recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint).

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Section: Linear Theorymentioning

confidence: 88%

“…The neutral modes are either odd or even but have no well-defined wave number. It follows that when neutral curves corresponding to modes of like parity (odd-odd, or even-even) cross, they generically reconnect [19], producing a quite different neutral curve topology. Only odd-even crossings remain structurally stable.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic snaking in bounded domains

Houghton

Knobloch

2009

Phys. Rev. E

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“…Ref. [32]. The figure shows the phase eigenvalue increasing rapidly from zero along the branch of odd parity states together with a more slowly increasing (negative) translation eigenvalue.…”

Section: Stabilitymentioning

confidence: 94%

Spatial localization in heterogeneous systems

2014

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We study spatial localization in the generalized Swift-Hohenberg equation with either quadratic-cubic or cubic-quintic nonlinearity subject to spatially heterogeneous forcing. Different types of forcing (sinusoidal or Gaussian) with different spatial scales are considered and the corresponding localized snaking structures are computed. The results indicate that spatial heterogeneity exerts a significant influence on the location of spatially localized structures in both parameter space and physical space, and on their stability properties. The results are expected to assist in the interpretation of experiments on localized structures where departures from spatial homogeneity are generally unavoidable.

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“…Such an O(2)-symmetry can also be present, e.g., in the Saltzman model (cf. [15]) and implies that the bifurcating traveling waves have a stronger steady state character with velocity predicted by the linear problem. However, the reflection symmetry is broken for differing viscosities, and we therefore rely on the center manifold reduction for Andronov-Hopf bifurcations.…”

Section: Introductionmentioning

confidence: 94%

Global Stability and Local Bifurcations in a Two-Fluid Model for Tokamak Plasma

Zhelyazov¹,

Han-Kwan²,

Rademacher³

2015

SIAM J. Appl. Dyn. Syst.

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We study a two-fluid description for high and low temperature components of the electron velocity distribution in an idealized tokamak plasma evolving on a cylindrical domain, taking into account nonlinear drift effects only. We refine previous results on the laminar steady state stability and include viscosity. Taking the temperature difference as the primary parameter, we show that linear instabilities and bifurcations occur within a finite interval and for small enough viscosity only, while the steady state is globally stable for parameters sufficiently far outside the interval. We find that primary instabilities always stem from the lowest spatial harmonics for aspect ratios of poloidal versus radial extent below some value larger than 2. Moreover, we show that any codimension-one bifurcation of the laminar state is supercritical, yielding spatio-temporal oscillations in the form of traveling waves, and hence locally stable for such bifurcations destabilizing the laminar state. In the degenerate case, where the instability region in the temperature difference is a point, these solutions form an arc connecting the bifurcation points. We also provide numerical simulations to illustrate and corroborate the analysis and find additional bifurcations of the traveling waves. Introduction.Existence of coherent states and their stability are fundamental questions in tokamak plasma theory and application; see, e.g., [3,7,10,16,17,18]. We study stability and bifurcations in a simple model for high and low temperature phases near the plasma edge, which appears in [20] and is a viscous variant of the model derived in [10]. It captures nonlinear effects due to electric drift in the magnetic field, the "E × B" drift, and the electron temperature gradient only. Our results highlight the role of these selected effects and give detailed information, which seems currently not possible for the much more accurate Vlasov-Maxwell or Vlasov-Poisson models considered in, e.g., [3,18].

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Mode Interactions in Large Aspect Ratio Convection

Cited by 28 publications

References 14 publications

Homoclinic snaking in bounded domains

Homoclinic snaking in bounded domains

Spatial localization in heterogeneous systems

Global Stability and Local Bifurcations in a Two-Fluid Model for Tokamak Plasma

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