Thursday, May 4, Clark Bldg., Room C238, 9am

Influence of distant boundaries on travelling-wave instabilities

Michael Proctor
Department of Applied Math and Theoretical Physics
University of Cambridge, UK

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 wavetrain into adjacent wavetrains with different frequencies, wavenumbers and amplitudes, separated by a front. The process of frequency selection is discussed in detail, and the selected frequency is shown to determine the wavenumber and amplitude of the wavetrains. The upstream boundary (with respect to the group velocity) plays a crucial part in fixing the frequency of the nonlinear wavetrain, and hence its stability properties. In contrast in the convectively unstable regime all perturbations decay, although persistent structures can be maintained by the addition of small amplitude noise.The upstream boundary, however distant, continues to play an essential role in frequency selection, with the result that the structures induced by noise are of universal form. A general theory is developed that predicts the selected frequency and wavenumber for both primary and secondary convective instabilities and the results are illustrated using the complex Ginzburg-Landau equation and a mean-field dynamo model of magnetic field generation in the Sun.
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