Noise-sustained structures due to convective instability in finite domains

In large aspect ratio systems with broken reflection symmetry the onset of instability is closely related to the threshold for absolute instability in the corresponding unbounded system. The upstream boundary (with respect to the group velocity) plays a crucial part in fixing the frequency of the nonlinear wavetrain that results, 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 wave number for both primary and secondary convective instabilities, and the results illustrated using the complex Ginzburg–Landau equation and a mean-field dynamo model of magnetic field generation in the Sun.