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On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

Published online by Cambridge University Press:  10 December 2019

Shane G. Walsh Open the ORCID record for Shane G. Walsh [Opens in a new window]  and
Miguel D. Bustamante Open the ORCID record for Miguel D. Bustamante [Opens in a new window]
Shane G. Walsh*
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin4, Ireland
Miguel D. Bustamante
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin4, Ireland
*
†Email address for correspondence: shanewalsh11235@gmail.com
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Abstract

We study numerically the region of convergence of the normal form transformation for the case of the Charney–Hasagawa–Mima (CHM) equation to investigate whether certain finite amplitude effects can be described in normal form coordinates. Specifically, we consider a Galerkin truncation of the CHM equation to four Fourier modes, and ask: Can the finite amplitude phenomenon known as precession resonance (Bustamante et al., Phys. Rev. Lett., vol. 113, 2014, 084502) be described by the normal form variables of wave turbulence theory? The answer is no, due to the failure of convergence of the normal form transformation at the wave amplitudes required for precession resonance. We show this by first characterising precession resonance in the original system, then, searching for this resonance in the normal form equations of motion. We find a large discrepancy between the dynamics of the original system and the transformed normal form system. This prompts an investigation into the convergence of the transformation in the state space of wave amplitudes. We find that the numerically calculated region of convergence is a region of finite wave amplitudes, well beyond the usual limit of weak nonlinearity. However, the amplitudes at the boundary of this region of convergence (i.e. the amplitudes at which the normal form transformation begins to diverge) match closely with the amplitudes at which precession resonance occurs in the original system. We conclude that the precession resonance mechanism cannot be explained by the usual methods of normal forms in wave turbulence theory, so a more general theory for intermediate nonlinearity is required.

JFM classification

Turbulent Flows: Wave-turbulence interactions Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 884 , 10 February 2020 , A28
Copyright
© 2019 Cambridge University Press

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