Patterns at Low Inclination Angles

The last subsection of this section, deals with patterns, that occur at very low inclination angles. These are often similar to patterns that exist in horizontal forced convection and that are discussed in detail in the next chapter. Several of these patterns coexist at the same time in different areas of the cell.

For small but finite inclination angles (5°< γ <10°) varicose pattern (VP) coexist withcoherent kink lines(KL). A very similar state were already found by McCoy and is described in detail in [69]. One kink is a localized phase jump of 2π (Fig. 4.18a). These phase jumps are aligned on a line, which often spans over the whole cell. At onset of KL, each longitudinal roll has only one such phase kink. The kink line is mostly straight, but tend to bend towards the sidewalls.

With increasing ε more kink lines occur and are getting closer to each other. At sufficient highε, kink lines cluster together and form patches of oblique rolls and of subharmonic resonances (Fig. 4.18b). Often varicose pattern form between adjacent kink lines (Fig. 4.18d). The kink lines drift slowly in downhill direction (movie available on enclosed CD-ROM)with a velocity of ≈ 0.7d/τ_t (for γ = 5° and ε = 2.42). The drift velocity increases with increasing shear flow (increase inε and γ).

The hysteretic behavior for varicose pattern and kink lines is different. In the horizontal case and at ε at which VP exist, a reduction of ε results in a reduced wiggling amplitude of VP in the same way as this amplitude grows with increasing ε. On the other hand reducing εfrom a spatio-temporal chaotic regime as in Fig.

4.8 or a kink state cannot reproduce VP anymore. Instead, straight longitudinal rolls form at some places of the convection cell, while in other regions kinks and kink clusters still exist (Fig. 4.18c). By reducing ε the size of the kink cluster shrinks and finally disappears completely at much lower ε (e.g., ε= 1.40 at γ = 0°) as the instability of straight rolls occurred due to an increase of ε (ε = 2.65 at γ = 0°).

Atγ = 10°one observes atε= 2.16 the onset of a varicose pattern. This varicose pattern is no longer steady in time, but drifts in uphill direction (movie available on enclosed CD-ROM). With increasing ε one notices that the amplitude of the phase modulation is not constant over the whole cell anymore (see Fig. 4.19). In some areas the wickling - the amplitude of the phase modulation - is very strong

a b

c d

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Figure 4.18: Single kink line at γ = 5° andε= 2.42 (a). Subharmonic resonances at γ = 5° and ε= 3.16 (b). Kink cluster at γ = 0° and ε = 1.69 (c). Often, VP form in between adjacent kinks lines (d). The red stripes on top of each image mark the location of the forcing structure.

4.2 Longitudinal Forcing 97

while in other areas the rolls are almost straight.

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Figure 4.19: Varicose pattern at γ = 10° and ε = 2.31. (a) shows a shadowgraph image and (b) is the corresponding phase field. Green encircled areas mark areas, where the amplitude of the phase modulation is reduced. The red stripes on top of each image mark the location of the forcing texture.

Atε= 2.67 neighboring rolls merge locally and produce kink defects (Fig. 4.20a).

However, these kinks are neither very sharp steps in phase, nor are they aligned along a very straight line. Instead, they are only loose correlated in space and chaotic in time. This state looks like a mixture of varicose pattern, kink lines and crawling rolls. These crawling kinks (CK) are shown in Fig. 4.20b. They were observed only for inclination of 10° and seem to be a mixed chaotic state, so that they are not marked in the phase diagram in Fig. 4.3.

Very interesting are these patterns if one lowers the control parameter and ob-serves the evolution of CK. Here again a hysteretic behavior is observed. The pattern does not turn back to VP if one lowers ε. Instead, one observes small areas of size ≈ 3λf in y-direction and various length in x-direction where a dy-namic state exist (movie available on enclosed CD-ROM). There, dydy-namic crawl-ing kinks with a subharmonic character were observed, while in the rest of the cell steady longitudinal rolls with wave vector~qf existed (see Fig. 4.21and movie on enclosed CD-ROM). One even can observe these localized crawling kinks for ε= 1.67. For comparison, in experiments were ∆T was adiabatically increased, forced longitudinal rolls were stable up to ε= 2.65.

The fact that in a bistable regime a localized state is found is not unique to the system, presented here. Similar localized states were also found in other pattern forming systems. In binary convection localized traveling waves were

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Figure 4.20: Crawling kinks and varicose pattern at ε = 2.67 (a). Fully chaotic crawling kinks at ε= 2.90 (b). Both images are taken at γ = 10° and show the whole cell. The red stripes on top of the left image mark the forcing texture.

found embedded in on otherwise quiescent fluid, when the Rayleigh number was slightly higher then the critical one [89]. In Taylor Couette flow of dilute polymer solution for example localized vortex pairs where found when the rotation rate were reduced slowly from a state were chaotically oscillating vortices exists, to a simple Couette shear flow [90]. Another example are localized spikes in ferro fluids [90]. These occur when the fluid shows Rosensweig spikes and the magnetic field is slowly reduced.

4.3 Transverse Forcing 99

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Figure 4.21: Localized crawling kinks at ε= 1.72 and γ = 10°. The image show the whole cell. The red stripes mark the forcing texture.

4.3 Forcing with a Wave Vector Parallel to