Rossby waves are the slow atmospheric waves that propagate thousands of kilometres on the time scale of days and are associated with weather. The small amplitude linear theory of Rossby waves on the sphere goes back over two centuries to Laplace’s tidal equations and is today considered thoroughly understood. However, with a more realistic background flow that includes both the tropical trade winds and the midlatitude jetstream, the global wave theory has not been fully established. To study this problem, this thesis uses the rotating shallow water (RSW) equations on the sphere as a model for Earth’s atmosphere. The spectrum of the RSW equations is numerically computed using a Galerkin method. It is found that the Rossby spectrum of this global model consists of two parts that naturally correspond respectively to local tropical and midlatitude theories. The first part of the spectrum is the countably infinite set of discrete eigenmodes with arbitrarily small wavelength near the equator, consistent with the local tropical β-plane theory. These discrete modes however achieve only a finite limiting wavelength in the midlatitudes. To account for smaller scales in the midlatitudes, it is necessary to consider the continuous spectrum that results from regular singular points in the RSW operator arising from shear in the background flow. The regular singular points correspond to critical latitudes, which prevent wave propagation from the midlatitudes into the tropics. The numerical results are complemented by small wavelength asymptotics of ray theory and Wentzel–Kramers–Brillouin (WKB) analysis to gain an understanding of the local wave- length, amplitude and group velocity for Rossby waves. The global understanding of Rossby waves presented in this thesis is used to provide some explanation of the small number of Rossby modes found in long-term climatological observations of theatmosphere.
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Thesis advisor: Muraki, David
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