Convective and absolute instabilities of fluid flows in finite geometry

Abstract

Dynamics of linear and nonlinear waves in driven dissipative systems in finite domains are considered. In many cases (for example, due to rotation) the waves travel preferentially in one direction. Such waves cannot be reflected from boundaries. As a consequence in the convectively unstable regime the waves ultimately decay; only when the threshold for absolute instability is exceeded can the waves be maintained against dissipation at the boundary. Secondary absolute instabilities are associated with the break-up of a wave train into adjacent wave trains with different frequencies, wave numbers and amplitudes, separated by a front. The process of frequency selection is discussed in detail, and the selected frequency is shown to determine the wave number and amplitude of the wave trains. The results are described using the complex Ginzburg-Landau equation and illustrated using a mean-field dynamo model of magnetic field generation in the Sun.