Results of Ginzburg-Landau equation

The large aspect-ratios, which can be achieved comparatively easily in EHC, make this system particularly well suited to test predictions of GLEs. Experimental results for the existence and stability of normal roll patterns in the q-e plane (Busse balloon) are shown in Fig. 13.7 [28, 36]. Similar experiments were reported by Ref [29]. The symbols give the experimental points and the curves represent parabolic fits for the neutral curve (N) and the 

Eckhaus stability limit (E). The secondary and tertiary instabilities that limit the regions from above will be discussed further below. 

The wave number q can be controlled within certain limits on the small wave number side by the so-called frequency-jump technique [32]. Since qc increases with external frequency one can prepare first a state for the desired wave number by choosing the appropriate frequency and then by jumping to the frequency and values. In the course of the experiment can still be varied. Outside of the neutral curve the rolls decay whereas inside they grow up to their non-linear saturation. The neutral point is then determined by extrapolation. In this way also the GL relaxation time t was determined. 

Slightly above onset the pattern should be dominated by the critical mode at q = qc and according to the weakly non-linear analysis (Section 13.3) the pattern amplitude should grow proportional to /e. This behaviour has been confirmed in experiments, where the optical contrast of the roll pattern was monitored as function of [59] using the shadowgraph method [34, 38, 99]. The proportionality factor, which is determined by the non-linear (cubic) coefficient g in the GLE, agreed satisfactorily with theoretical results [12]. 

In a next step the Eckhaus instability boundaries (curves E in Fig. 13.7), which probe non-linear aspects of the system, could be identified by observing the destabilization of the pattern via longitudinal modulations after a frequency jump [29, 36, 37] . Subsequently the 

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