Homoclinic snaking of localized states in doubly diffusive convection

Beaume, Cédric; Bergeon, Alain; Knobloch, Edgar

doi:10.1063/1.3626405

Cited by 36 publications

(43 citation statements)

References 51 publications

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“…These branches are expected to bifurcate from the first branch of periodic states at a larger amplitude than the 1-pulse solutions and to snake in the same region as the 1-pulse states. 28,38 However, the connection between the 1-and 2-pulse branches observed here has not been observed in other systems. We surmise that it forms as a result of a tangency between these branches that forms as β decreases.…”

Section: Nonlinear Resultsmentioning

confidence: 41%

“…As a result slanted snaking is absent, and the convectons instead undergo standard homoclinic snaking in a well-defined Rayleigh number interval much as in other problems with no-slip boundary conditions. 8,28,38,39 In particular, we no longer expect to find convectons below the saddle-node of the periodic states or in the supercritical regime, and no such solutions have been found.…”

Section: Discussionmentioning

confidence: 99%

“…40,41 More generally, multipulse states are found in the snaking region of 1-pulse states, both in model equations like the Swift-Hohenberg equation 42 and in convection problems without a conserved quantity. 28 In these systems, however, the 1-pulse and 2-pulse states remain distinct, and no connections between them are present. Thus, the observation in the present paper of the termination of 1-pulse states on a branch of 2-pulse states is to our knowledge new.…”

Section: Discussionmentioning

confidence: 99%

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Localized rotating convection with no-slip boundary conditions

et al. 2013

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Localized patches of stationary convection embedded in a background conduction state are called convectons. Multiple states of this type have recently been found in two-dimensional Boussinesq convection in a horizontal fluid layer with stress-free boundary conditions at top and bottom, and rotating about the vertical. The convectons differ in their lengths and in the strength of the self-generated shear within which they are embedded, and exhibit slanted snaking. We use homotopic continuation of the boundary conditions to show that similar structures exist in the presence of no-slip boundary conditions at the top and bottom of the layer and show that such structures exhibit standard snaking. The homotopic continuation allows us to study the transformation from slanted snaking characteristic of systems with a conserved quantity, here the zonal momentum, to standard snaking characteristic of systems with no conserved quantity. C 2013 AIP Publishing LLC. [http://dx

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Section: Nonlinear Resultsmentioning

confidence: 41%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Localized rotating convection with no-slip boundary conditions

et al. 2013

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“…The former enhances concentration pumping 39 while the latter suppresses pumping. 22 Residual concentration pumping is, therefore, likely to be present and its presence will reduce the background concentration gradient sensed by the convecton. We expect this effect to be proportional to the square of the amplitude of convection and hence linear in the Rayleigh number.…”

Section: Discussionmentioning

confidence: 99%

“…Since then stationary localized convection has been extensively studied in two-dimensional (2D) doubly diffusive convection in a horizontal layer, both with Soret effect [18][19][20] and without. 21,22 Solutions of this type, hereafter referred to as convectons, may be viewed as homoclinic orbits in space connecting the conduction state to itself and are associated with heteroclinic orbits or fronts connecting the conduction state to a periodic roll state and back again. These solutions have recently been computed in natural three-dimensional (3D) doubly diffusive convection in a vertically extended cavity 23 and are accompanied by secondary instabilities leading to the twisting of some of the convection rolls, an instability only allowed by the additional freedom provided by the third dimension.…”

Section: Introductionmentioning

confidence: 99%

Nonsnaking doubly diffusive convectons and the twist instability

2013

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Doubly diffusive convection in a three-dimensional horizontally extended domain with a square cross section in the vertical is considered. The fluid motion is driven by horizontal temperature and concentration differences in the transverse direction. When the buoyancy ratio N = −1 and the Rayleigh number is increased the conduction state loses stability to a subcritical, almost two-dimensional roll structure localized in the longitudinal direction. This structure exhibits abrupt growth in length near a particular value of the Rayleigh number but does not snake. Prior to this filling transition the structure becomes unstable to a secondary twist instability generating a pair of stationary, spatially localized zigzag states. In contrast to the primary branch these states snake as they grow in extent and eventually fill the whole domain. The origin of the twist instability and the properties of the resulting localized structures are investigated for both periodic and no-slip boundary conditions in the extended direction. C 2013 AIP Publishing LLC. [http://dx

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Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

Tuckerman¹,

Langham²,

Willis

2018

Computational Methods in Applied Sciences

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Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.

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Homoclinic snaking of localized states in doubly diffusive convection

Cited by 36 publications

References 51 publications

Localized rotating convection with no-slip boundary conditions

Localized rotating convection with no-slip boundary conditions

Nonsnaking doubly diffusive convectons and the twist instability

Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

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