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Article
Article Type:
Research Article

Linear Stability of Modons on a Sphere

E. C. Neven Atmospheric Dynamics Group, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom, and Vortex Dynamics Group, Fluid Dynamics Laboratory, Eindhoven University of Technology, Eindhoven, Netherlands

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Print Publication:
01 Aug 2001
DOI:
https://doi.org/10.1175/1520-0469(2001)058<2280:LSOMOA>2.0.CO;2
Page(s):
2280–2305
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Abstract

The linear stability of two stationary dipolar modon solutions of the nondivergent barotropic vorticity equation on a rotating sphere is investigated. A numerical normal mode analysis of the linearized equation is performed by solving the eigenvalue problem in a spectral model. The modons are models of observed vortical structures in the atmosphere such as planetary waves and atmospheric blocks. The wavelike modon in an eastward zonal flow represents the class of partially localized vortical waves with phase velocities inside the Rossby wave regime. The localized modon in a westward zonal flow represents the class of desingularized point vortex pairs with phase velocities outside the Rossby wave regime. The research into the stability properties of these modons may lead to a deeper insight into the decay and persistency of planetary waves and atmospheric blocks and thereby into the low-frequency variability of the atmosphere.

The convergence of the modes with resolution is studied with the matrix method for each resolution T10–T100 and with the iterative power method for every five resolutions T80–T170 and for T341. The modes are tracked through the resolutions using correlations between eigenvectors at subsequent resolutions. The four most unstable modes of the wavelike modon are convergent over T22. For the localized modon, the most unstable mode is convergent over T66, whereas other modes appear to converge to zero growth rate over T100. Both modons are unstable with an e-folding time of several days.

The structure of the modes is studied in detail for resolution T85. The most unstable modes have their largest local amplitudes in regions with the strongest gradients of potential vorticity. The modes propagate around the amphidromic points, local centers of revolution, which characterize the topological structure of the modes. The most unstable mode of the wavelike modon has a tripolar structure and propagates unhindered across the boundary circle between the inner and outer region of the modon. The most unstable mode of the localized modon has a quadrupolar structure and propagates north and south of the boundary circle, which is a critical line and an impenetrable barrier. The modons are representatives of two different classes of vortical structures and their quite distinct mode structures can be understood using the refractive index interpretation.

The Eliassen–Palm theorem applied to modes as perturbations on modons leads to a condition on the mean spectral wavenumber for an unstable mode: the barotropic wedge of instability in the diagram of perturbation relative vorticity versus perturbation streamfunction. A causal argument regarding the time required for an instability to radiate propagating waves leads to the phase speed condition for a global propagating mode: the spatial extent of the mode is related to the ratio of its e-folding time and oscillation period. These theoretical results are supported by high-resolution numerical results: wavelike modons have localized (trapped) modes, whereas localized modons have all but one wavelike (radiating) mode. The localized modon has a single isolated unstable mode with an additional continuum of propagating neutral modes. For both modons, the most unstable mode propagates eastward between the cells in the inner region and westward along the boundary circle in the outer region.

Corresponding author address: Dr. E. C. Neven, Vortex Dynamics Group, Fluid Dynamics Laboratory, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands. Email: e.c.neven@tue.nl

Abstract

The linear stability of two stationary dipolar modon solutions of the nondivergent barotropic vorticity equation on a rotating sphere is investigated. A numerical normal mode analysis of the linearized equation is performed by solving the eigenvalue problem in a spectral model. The modons are models of observed vortical structures in the atmosphere such as planetary waves and atmospheric blocks. The wavelike modon in an eastward zonal flow represents the class of partially localized vortical waves with phase velocities inside the Rossby wave regime. The localized modon in a westward zonal flow represents the class of desingularized point vortex pairs with phase velocities outside the Rossby wave regime. The research into the stability properties of these modons may lead to a deeper insight into the decay and persistency of planetary waves and atmospheric blocks and thereby into the low-frequency variability of the atmosphere.

The convergence of the modes with resolution is studied with the matrix method for each resolution T10–T100 and with the iterative power method for every five resolutions T80–T170 and for T341. The modes are tracked through the resolutions using correlations between eigenvectors at subsequent resolutions. The four most unstable modes of the wavelike modon are convergent over T22. For the localized modon, the most unstable mode is convergent over T66, whereas other modes appear to converge to zero growth rate over T100. Both modons are unstable with an e-folding time of several days.

The structure of the modes is studied in detail for resolution T85. The most unstable modes have their largest local amplitudes in regions with the strongest gradients of potential vorticity. The modes propagate around the amphidromic points, local centers of revolution, which characterize the topological structure of the modes. The most unstable mode of the wavelike modon has a tripolar structure and propagates unhindered across the boundary circle between the inner and outer region of the modon. The most unstable mode of the localized modon has a quadrupolar structure and propagates north and south of the boundary circle, which is a critical line and an impenetrable barrier. The modons are representatives of two different classes of vortical structures and their quite distinct mode structures can be understood using the refractive index interpretation.

The Eliassen–Palm theorem applied to modes as perturbations on modons leads to a condition on the mean spectral wavenumber for an unstable mode: the barotropic wedge of instability in the diagram of perturbation relative vorticity versus perturbation streamfunction. A causal argument regarding the time required for an instability to radiate propagating waves leads to the phase speed condition for a global propagating mode: the spatial extent of the mode is related to the ratio of its e-folding time and oscillation period. These theoretical results are supported by high-resolution numerical results: wavelike modons have localized (trapped) modes, whereas localized modons have all but one wavelike (radiating) mode. The localized modon has a single isolated unstable mode with an additional continuum of propagating neutral modes. For both modons, the most unstable mode propagates eastward between the cells in the inner region and westward along the boundary circle in the outer region.

Corresponding author address: Dr. E. C. Neven, Vortex Dynamics Group, Fluid Dynamics Laboratory, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands. Email: e.c.neven@tue.nl

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