Simulations at large P

In this section, we analyze the patterns at large P obtained in various simulations at different parameter sets which have been indicated in Fig. 2.44. As already known from Fig. 2.14, stable zonal rolls exist in the entireεo, τ regime left to the dashed line.

This is confirmed by simulations started with random initial conditions. Indeed, as shown in Fig. 2.45, zonal roll structures perfectly aligned with the magnetic field are obtained after a relatively short time.

To confirm the existence of locally stable oblique rolls for small τ (see also Fig.

2.16) value we disturbed a χ = 45^o pattern obtained from a Galerkin solution. At first, we convince ourselves that superimposed noise in Fourier space would quickly die out. Alternatively, we applied a localized perturbation by reducing the convection in a small patch in the position space (see the second and third patterns in Fig. 2.46).

In this case χ = 45^o rolls changed their orientation into χ = 63^o oblique roll which, according to Fig. 2.16, are stable.

In the case of moderate τ values which are comparable to the value of Q oblique rolls, as predicted by linear analysis already at onset, are indeed found for low values of εo. A typical time evolution from random initial conditions to an oblique roll pattern is shown in Fig. 2.47. A more detailed analysis of the stable regions of oblique rolls has been presented in Sec. 2.3.2. The preferred angle χ of oblique rolls increases as εo is increased. In particular we have found a large regime of stable zonal rolls (χ= 90^o) to the right of the dashed SW line in Fig. 2.44. Note that the range of stable zonal rolls in Fig. 2.17 overlaps to considerable extent with the region τ > τL where the stable critical oblique rolls are found. Accordingly we expect hysteresis phenomena. This tendency towards zonal rolls with increasing εo has also been observed experimentally [28] as well as theoretically [25] in the case Q= 0.

To the right of the dashed line of Fig. 2.14a at values of εo of order unity the for-mation of the stable zonal roll depends on the initial conditions. With superimposed noise, this property is demonstrated in Fig. 2.48 where oblique rolls evolve into a pat-tern of zonal rolls with two dislocations which are likely to disappear after much longer integration times. In contrast, when the time integration is started with random initial conditions a stationary pattern of zonal rolls has not been obtained asymptotically.

Instead a spatio-temporally chaotic pattern persists in which zonal rolls dominate only locally as demonstrated in Fig. 2.49. Another example of this phenomenon is demon-strated in Figs. 2.50 and 2.51 for a much higher value of τ. A slightly disturbed

2 4 6 8 10 12 14 16 18

τ

0 0.5 1 1.5 2

ε

neutral curve shortwave inst.

x x

Figure 2.44: Stability diagram (see also Fig. 2.14) in the τ −ε_o plane for Q = 14 and P = 10. Simulations discussed in the text correspond to the different symbols in the figure. Filled circles denote the oblique patterns, open squares indicate zig-zag patterns, open triangles correspond to knot patterns, filled diamonds indicate spatio-temporally chaotic convection and stars indicate the points where competition between different patterns has been observed.

Figure 2.45: Zonal rolls for τ = 3, Q = 14, εo = 0.8. The plots from left to right correspond to the times t= 0, 14.3, 28.57 and 114.3 in units oftv (see the filled square in Fig. 2.44). The aspect ratio is Γ = 20. The x−coordinate increases towards the right and the y−coordinate is directed upwards.

zonal roll pattern recovers after some time (see Fig. 2.50), while starting from initial conditions we arrive at the “chaotic” attractor (Fig. 2.51).

The co-existence of time independent attractors andspatio-temporally chaoticones is a well known phenomenon in problems of pattern formation. For an example we

Figure 2.46: Stable oblique rolls (χ = 45^o) for τ = 2, Q= 14, εo = 0.31 (see also Fig.

2.16). The plots from left to right correspond to the times t= 0.5, 15.3, 19.8 and 31.3 (see the filled circle in Fig. 2.44). The aspect ratio is Γ = 20. Here and in the following figures the temperature field in the plane z = 0.5 is shown.

Figure 2.47: Oblique rolls for τ = 8.7, Q = 14, εo = 0.18 (see the filled circle in Fig.

2.44). The plots from left to right correspond to the timest= 0, 6.3, 12.6 and 50. The aspect ratio is Γ = 28.

Figure 2.48: Time evolution for τ = 9.1, Q = 14, εo = 0.65 starting from initial conditions of oblique rolls with superimposed noise (see the star in Fig. 2.44). The plots from left to right correspond to the times t = 0.7, 3.3, 5.8 and 42.8. The aspect ratio is Γ = 22.

refer to the case of spiral defect chaos in RB convection [33, 34, 35].

For relatively higher values of τ (> τU) the axial rolls become unstable against the subharmonic varicose (SHV) instability according to the Galerkin analysis. ButSHV patterns are only stable for fairly small εo (see Sec. 2.4.2). Thus a stationary SHV pattern evolves to the oblique roll patterns at larger εo. An example is shown in Fig.

2.52. Note that SHV instability is well expressed in the transients; the angle of the

Figure 2.49: Time evolution for the same parameters as in Fig. 2.48 starting from random initial conditions (see the star in Fig. 2.44). The plots from left to right correspond to the times t = 4.83, 7.17, 11.3 and 46.7, respectively. Time is in units of t_v. The aspect ratio is Γ = 22.

Figure 2.50: Stable zonal rolls forτ = 14,Q= 14, εo = 1.54, qzonal = 4 (see the star in Fig. 2.44). The plots from left to right correspond to times t = 0, 0.125, 0.25 and 5.

The aspect ratio is Γ = 20.

Figure 2.51: Time sequence for the random initial conditions for the same parameters as in Fig. 2.50 (see the star in Fig. 2.44). The plots from left to right correspond to times t= 0, 0.44, 1.32 and 22. The aspect ratio is Γ = 22.

oblique rolls is about χ = 60^o characteristic for hexarolls which are a special case of SHV instability.

In contrast to Fig. 2.52 the system also exploits the knot instability which bifurcates nearby to arrive at the stable oblique rolls as shown in Fig. 2.53. The knots first become strong in zonal direction, then they loose their original orientation and become like oblique rolls. After some intermediate steps which include some phase shifts of

Figure 2.52: Stable oblique rolls just above the instability line for τ = 10, Q = 14, εo = 0.19 (see the filled circle in Fig. 2.44). The plots from left to right correspond to times t= 0, 1.1, 1.27, 1.81, 2, 2.9, 3.6 and 65.5. The aspect ratio is Γ = 20.

the rolls (see the patterns in middle line) the pattern evolves into an oblique structure.

The transition interval of the shifting process can also be seen in the time variation of two leading modes (Fig. 2.54). Here, A and B amplitudes show strong fluctuations during the transition time.

At slightly higher value of εo a chaotic pattern is found which is dominated by a switching back and forth between the two equivalent oblique roll systems with angles of ψ =±37^o as shown in Fig. 2.55. For even larger values of εo the oblique rolls are replaced by fully phase-turbulent convection in this regime. An example is shown in Fig. 2.56. We have included a graph of the power spectrum which corresponds to the last spatial pattern in Fig. 2.56. Note that the ring- like wave vector distribution is nearly isotropic with the approximate mean radius given by|qc |. This feature can be understood on the basis of the property that the stabilizing effects exerted by rotation and by the magnetic field in the two orthogonal directions are approximately equal.

In a region surrounding the open squares in Fig. 2.44 stationary zig- zag patterns have been found as attractors as shown, for example, in Fig. 2.57. Outside this region, however, spatio-temporally chaotic convection is usually found when the simulations are started with random initial conditions.