Barotropic Waves in Straight Parallel Flow with Curved Velocity Profile

Abstract

The paper contains a study of the dynamics of two-dimensional straight parallel flow with a continuously curved profile. The differential equation for the stream-function of a neutral wave disturbance is used as an analytical tool to investigate the properties and the evolution of waves having a certain initial configuration of the flow. The behaviour of the waves is explained by using an extension of the method applied by Holmboe (1953) in his study of the Kelvin and Rayleigh waves. In order to do that, the problem has been approached from a point of view which deviates somewhat from the customary treatment. The method is first illustrated by a model with a hyperbolic tangent profile (Th-flow), which is shown to have a very simple analytical solution for the stationary wave L8. The form of the solution makes it possible to interpret it for other wave lengths (L?L8) as a forced stationary wave which is the resultant of a non stationary physical wave and a Rayleigh wave induced by the proper infinite sliding vorticities at the central level. When this foreign field is removed from the solution, the remaining field, which now is a physically possible initial configuration of the flow, will propagate with different speed from level to level. As the tilt develops the wave will begin to amplify if L > L8, and it will be damped if L < L8. So the stationary wave represents the transition from short stability waves to long instability waves as had been anticipated by Fjørtoft (1). The method is then extended to more general types of profiles. Some general results on the conditions for the existence of stationary waves, which were proved or anticipated by Rayleigh (1880), Fjørtoft (1950) and Höiland (1951), are given a unified treatment. It is finally shown that, if a system has stationary waves, they represent the transition from damped to amplified waves. If the number of stationary waves is even, both the shortest and the longest waves are damped waves, and the intermediate intervals of the spectrum are alternate bands of amplified and damped Iwaves. If the number of stationary waves is odd, the longest waves in the spectrum are amplified.