### Abstract

A weakly nonlinear stability theory is developed for a rotating flow confined in a cylinder of elliptic cross-section. The straining field associated with elliptic deformation of the cross-section breaks the SO(2)-symmetry of the basic flow and amplifies a pair of Kelvin waves whose azimuthal wavenumbers are separated by 2, being referred to as the Moore-Saffman-Tsai-Widnall (MSTW) instability. The Eulerian approach is unable to fully determine the mean flow induced by nonlinear interaction of the Kelvin waves. We establish a general framework for deriving the mean flow by a restriction to isovortical disturbances with use of the Lagrangian variables and put it on the ground of the generalized Lagrangian-mean theory. The resulting formula reveals enhancement of mass transport in regions dominated by the vorticity of the basic flow. With the mean flow at hand, we derive unambiguously the weakly nonlinear amplitude equations to third order for a nonstationary mode. By an appropriate normalization of the amplitude, the resulting equations are made Hamiltonian systems of four degrees of freedom, possibly with three first integrals identifiable as the wave energy and the mean flow.

Original language | English |
---|---|

Article number | 014042 |

Journal | Physica Scripta |

Volume | 88 |

Issue number | T155 |

DOIs | |

Publication status | Published - Jul 1 2013 |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics
- Mathematical Physics
- Condensed Matter Physics

### Cite this

*Physica Scripta*,

*88*(T155), [014042]. https://doi.org/10.1088/0031-8949/2013/T155/014042

**Hamiltonian bifurcation theory for a rotating flow subject to elliptic straining field.** / Fukumoto, Y.; Mie, Y.

Research output: Contribution to journal › Article

*Physica Scripta*, vol. 88, no. T155, 014042. https://doi.org/10.1088/0031-8949/2013/T155/014042

}

TY - JOUR

T1 - Hamiltonian bifurcation theory for a rotating flow subject to elliptic straining field

AU - Fukumoto, Y.

AU - Mie, Y.

PY - 2013/7/1

Y1 - 2013/7/1

N2 - A weakly nonlinear stability theory is developed for a rotating flow confined in a cylinder of elliptic cross-section. The straining field associated with elliptic deformation of the cross-section breaks the SO(2)-symmetry of the basic flow and amplifies a pair of Kelvin waves whose azimuthal wavenumbers are separated by 2, being referred to as the Moore-Saffman-Tsai-Widnall (MSTW) instability. The Eulerian approach is unable to fully determine the mean flow induced by nonlinear interaction of the Kelvin waves. We establish a general framework for deriving the mean flow by a restriction to isovortical disturbances with use of the Lagrangian variables and put it on the ground of the generalized Lagrangian-mean theory. The resulting formula reveals enhancement of mass transport in regions dominated by the vorticity of the basic flow. With the mean flow at hand, we derive unambiguously the weakly nonlinear amplitude equations to third order for a nonstationary mode. By an appropriate normalization of the amplitude, the resulting equations are made Hamiltonian systems of four degrees of freedom, possibly with three first integrals identifiable as the wave energy and the mean flow.

AB - A weakly nonlinear stability theory is developed for a rotating flow confined in a cylinder of elliptic cross-section. The straining field associated with elliptic deformation of the cross-section breaks the SO(2)-symmetry of the basic flow and amplifies a pair of Kelvin waves whose azimuthal wavenumbers are separated by 2, being referred to as the Moore-Saffman-Tsai-Widnall (MSTW) instability. The Eulerian approach is unable to fully determine the mean flow induced by nonlinear interaction of the Kelvin waves. We establish a general framework for deriving the mean flow by a restriction to isovortical disturbances with use of the Lagrangian variables and put it on the ground of the generalized Lagrangian-mean theory. The resulting formula reveals enhancement of mass transport in regions dominated by the vorticity of the basic flow. With the mean flow at hand, we derive unambiguously the weakly nonlinear amplitude equations to third order for a nonstationary mode. By an appropriate normalization of the amplitude, the resulting equations are made Hamiltonian systems of four degrees of freedom, possibly with three first integrals identifiable as the wave energy and the mean flow.

UR - http://www.scopus.com/inward/record.url?scp=84880387639&partnerID=8YFLogxK

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U2 - 10.1088/0031-8949/2013/T155/014042

DO - 10.1088/0031-8949/2013/T155/014042

M3 - Article

AN - SCOPUS:84880387639

VL - 88

JO - Physica Scripta

JF - Physica Scripta

SN - 0031-8949

IS - T155

M1 - 014042

ER -