Abstract

The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151-176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D4 ⋉ T2-equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435-1455). We also find new phenomena in the normal form for a Hopf bifurcation with D4 ⋉ T2 symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063-1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.

Original languageEnglish
Pages (from-to)109-136
Number of pages28
JournalPhysica D: Nonlinear Phenomena
Volume139
Issue number1-2
DOIs
Publication statusPublished - 1 May 2000

Fingerprint

convection
symmetry
nonlinearity
planforms
interactions
orbits
conduction
cycles
fluids

Keywords

  • 3D magnetoconvection
  • Drifting solutions
  • Mode interaction

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Condensed Matter Physics

Cite this

The 1:2 Hopf/steady-state mode interaction in three-dimensional magnetoconvection. / Dawes, J. H.P.

In: Physica D: Nonlinear Phenomena, Vol. 139, No. 1-2, 01.05.2000, p. 109-136.

Research output: Contribution to journalArticle

@article{b156297a196f4b0b9a8135972557b944,
title = "The 1:2 Hopf/steady-state mode interaction in three-dimensional magnetoconvection",
abstract = "The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151-176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D4 ⋉ T2-equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435-1455). We also find new phenomena in the normal form for a Hopf bifurcation with D4 ⋉ T2 symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063-1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.",
keywords = "3D magnetoconvection, Drifting solutions, Mode interaction",
author = "Dawes, {J. H.P.}",
year = "2000",
month = "5",
day = "1",
doi = "10.1016/S0167-2789(99)00210-9",
language = "English",
volume = "139",
pages = "109--136",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "1-2",

}

TY - JOUR

T1 - The 1:2 Hopf/steady-state mode interaction in three-dimensional magnetoconvection

AU - Dawes, J. H.P.

PY - 2000/5/1

Y1 - 2000/5/1

N2 - The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151-176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D4 ⋉ T2-equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435-1455). We also find new phenomena in the normal form for a Hopf bifurcation with D4 ⋉ T2 symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063-1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.

AB - The first analysis of a properly three-dimensional mode interaction between steady and oscillatory forms of convection with different preferred wavenumbers is presented. By varying the fluid parameters for Boussinesq magnetoconvection we locate a point where the conduction state is unstable to both steady and oscillatory motion simultaneously. We then construct and analyse the normal form. The complex transition between steady and oscillatory convection near onset can be explained: this extends and completes the work of Clune and Knobloch (Pattern selection in three-dimensional magnetoconvection, Physica D 74 (1994) 151-176). Selecting the most marginal wavenumbers in the problem ensures that the analysis is relevant to the behaviour which would be observed in an unbounded plane layer. The symmetries of the resulting D4 ⋉ T2-equivariant bifurcation problem play a large role in determining the bifurcation structure and explain the appearance of interesting phenomena such as drifting solutions (A.M. Rucklidge, M. Silber, Bifurcations of periodic orbits with spatio-temporal symmetries, Nonlinearity 11 (1998) 1435-1455). We also find new phenomena in the normal form for a Hopf bifurcation with D4 ⋉ T2 symmetry (M. Silber, E. Knobloch, Hopf bifurcation on a square lattice, Nonlinearity 4 (1991) 1063-1106). The introduction of weakly non-Boussinesq effects leads to qualitative changes in the dynamics near onset: different convection planforms are stabilised and chaotic heteroclinic cycling behaviour is observed.

KW - 3D magnetoconvection

KW - Drifting solutions

KW - Mode interaction

UR - http://www.scopus.com/inward/record.url?scp=0345849831&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(99)00210-9

DO - 10.1016/S0167-2789(99)00210-9

M3 - Article

VL - 139

SP - 109

EP - 136

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -