Weakly nonlinear saturation of stationary resonance of a rotating flow in an elliptic cylinder

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  • Mie, Yoichi
    Faculty of Mathematics, Kyushu University
  • Fukumoto, Yasuhide
    Faculty of Mathematics and Mathematical Research Center for Industrial Technology, Kyushu University

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Abstract

We address weakly nonlinear stability of a uniformly rotating flow confined in a cylinder of elliptic cross-section to three-dimensional disturbances. A Lagrangian approach is developed to derive unambiguously the drift current induced by nonlinear interaction of isovortical disturbances. This approach rescues the insufficiency inherent in the Eulerian approach and provides a direct path to reach the amplitude equations in the Hamiltonian normal form. The nonlinear effect saturates the stationary instability mode, and asymptotic form of its saturation amplitude is gained, in a tidy form, in the short-wavelength regime.

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