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On the stability and propagation of barotropic modons in slowly varying media

University of British Columbia

Two aspects of the theory of barotropic modons are examined in this thesis. First, sufficient neutral stability conditions are derived in the form of an integral constraint for westward and eastward-travelling modons. It is shown that eastward-travelling and westward-travelling modons are neutrally stable to perturbations in which the energy is contained mainly in spectral components with wavenumber magnitudes (|ƞ|) satisfying |ƞ|<κ and |ƞ|>κ, respectively, where κ is the modon wavenumber. These results imply that when κ/|ƞ|>1 the slope of the neutral stability curve proposed by McWilliams et al.(l98l) for eastward-travelling modons must begin to increase as κ/|ƞ| increases. The neutral stability condition is computed with mesoscale wavenumber eddy energy spectra representative of the atmosphere and ocean. Eastward-travelling atmospheric modons are neutrally stable to the observed seasonally- and annually-averaged atmospheric eddies. The neutral stability of westward-travelling atmospheric modons and oceanic modons cannot be inferred on the basis of the observed wavenumber eddy energy spectra for the atmosphere and ocean. Second, a leading order perturbation theory is developed to describe the propagation of barotropic modons in a slowly varying medium. Two problems are posed and solved. A perturbation solution is obtained describing the propagation of an eastward-travelling modon modulated by a weak bottom Ekman boundary layer. The results predict that the modon radius and translation speed decay exponentially and that the modon wavenumber increases exponentially, resulting in an exponential amplitude decay in the streamfunction and vorticity. These results agree with the numerical solution of McWilliams et al.(l98l). A leading order perturbation theory is also developed describing modon propagation over slowly varying topography. Nonlinear hyperbolic equations are derived to describe the evolution of the slowly varying modon radius, translation speed and wavenumber for arbitrary finite-amplitude topography. To leading order, the modon is unaffected by meridional gradients in topography. Analytical perturbation solutions for the modon radius, translation speed and wavenumber are obtained for small-amplitude topography. The perturbations take the form of westward and eastward-travelling transients and a stationary component proportional to the topography. The general solution is applied to ridge-like and escarpment-like topographic configurations.

Text

2010-06-24

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http://hdl.handle.net/2429/25977

Mathematics

University of British Columbia

Graduate