Phase Velocity and Structure Calculations for Unstable Quasi-Geostrophic Waves in a Simple Baroclinic Westerly Current

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Abstract
A simple numerical method is used to calculate the vertical structure and the phase velocity, c = cr + cii, for small, sinusoidal, quasi-geostrophic, unstable perturbations in a baroclinic zonal current which is independent of latitude and is bounded above by a barotropic and isothermal stratosphere. Values of c for waves of a given effective length are determined as complex eigenvalues by assuming trial values, integrating the perturbation equations numerically between 1000 mb and the tropopause, and evaluating residuals which measure the extent to which the boundary conditions are not satisfied. The residuals are then used to obtain new estimates, which under favorable conditions converge to stable values if the first guess is sufficiently close. Some progress has been made toward automating the selection of first guesses. For middle latitudes (25°–75°) and moderate-to-strong vertical shears, convergence can usually be obtained for the range of effective wavelengths in which ci is a substantial fraction of its maximum value. Each eigenvalue for c has ci > 0 and is paired with a conjugate value. In simple cases the value found for c seems to be the only eigenvalue in a large region of the complex plane above the axis of reals. Sample calculations are given for latitude 45°.

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