Effect of stratification on the frequency of bounded Rossby modes over a non-flat bottom |
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| Authors: | Giuseppe Colantuono Robert Erdélyi Michael S Ruderman |
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| Institution: | 1. Department of Physics and Astronomy , University of Sheffield , Hounsfield Road, Hicks Building, Sheffield S3 7RH , UK g.colantuono@sheffield.ac.uk;3. School of Mathematics and Statistics , University of Sheffield , Hounsfield Road, Hicks Building, Sheffield S3 7RH , UK |
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| Abstract: | This work attempts to express and analyze the challenges, induced by stratification, affecting the Rossby-topographic eigenmodes of a closed domain with a general uneven bottom of arbitrary shape filled with a uniform fluid in the unperturbed configuration. The modified eigenmodes have been computed analytically: stratification is introduced in the mathematical form of a perturbation of a homogeneous fluid over a non-flat bottom. The eigenmodes lose their barotropic character and differences appear in the dynamical fields (velocity and pressure) from upper to lower layer, as expected. Expressions for the baroclinic and ageostrophic velocity components due to the perturbation are given. The analysis is carried out in the frame of linear shallow water approximation. All terms have been retained apart from nonlinear advection in the governing equations. We find that the frequencies of the eigenmodes change; an analytical expression of frequency correction as a function of layer density difference and interface depth is found. Initial results for some elementary geometrical settings with a waveguide bottom are determined and expressed in a concise, easily readable closed form. The results obtained in the shallow water approximation are expanded in series with respect to the Rossby number. Next, they are compared with the frequency correction obtained in an alternative framework in which the quasi-geostrophic approximation is used, and a purely baroclinic perturbation is imposed from the outset as the result of the introduction of stratification in the otherwise homogeneous fluid. In this scenario, reduced gravity and the ratio of upper to lower layer depth are, in turn, used as the expansion parameters in lieu of the Rossby number. |
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| Keywords: | Rossby waves Frequency perturbation Stratification Vertical modes: Bessel functions |
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