Abstract
The quasi-linear and non-linear growth of disturbances induced in a stably stratified layer with hyperbolic tangent shear is systematically examined based on shape assumption and by direct integrations. By the calculations based on the shape assumption, we obtain the maximum values of the velocity fluctuations, the time needed for the disturbances to attain the maxima and the averaged growth rate for various Richardson numbers and wavenumbers. The averaged growth rate offers somewhat smaller value than that derived by the direct integrations. But the difference is not significant. Comparison with the wind tunnel experiment shows rough justification for the shape assumption.
Development of Kelvin-Helmholtz waves is examined by direct numerical integrations. The time evolutions of the stream function, the velocity and the potential temperature are obtained by a non-linear system which includes five harmonic components. At early stages of the time evolution, there are little differences among non-linear systems of any degree including the quasilinear one because the higher harmonics are too small to affect the fundamental mode. But with lapse of time, discrepancies of wave pattern among the systems become significant. Namely in the non-linear systems of relatively lower degree including the quasi-linear system, periodic behavior is predominant while in the higher degree systems, such a behavior tends to disappear.
Some characteristic features of Kelvin-Helmholtz instability are discussed based on the results obtained by the numerical integrations. Considerations about reduction mechanism of Richardson number in the atmospheric planetary boundary layer, reradiation of internal gravity waves by Kelvin-Helmholtz instability and decay processes of Kelvin-Helmholtz billows are also proposed with reasonable speculations.
