The Eckhaus instability in hexagonal patterns

Mean flow in hexagonal convection: stability and nonlinear dynamics

“…In view of the effect of the mean flow it is worth noting that the regime in which the longitudinal mode is the relevant destabilizing mode is typically extremely small (below dashed-dotted line in fig.3) and it is very difficult to investigate that instability experimentally. In fact, even in numerical simulations of (4) (with β = 0) we found it difficult to separate the dynamics of the two modes (see also [29]). Note that due to the scaling of the quadratic coefficient in (4,5) the usually small range over which hexagons can be observed extends from µ ≈ 0 to µ ≈ 1.5. of figure 4a,b for small µ.…”

“…As discussed below (see also sec.4), the decomposition into transverse and longitudinal mode does not hold beyond the long-wave limit. It is replaced by a separation into a 'wide-splitting' and a 'narrow-splitting' mode [29]. The dashed-dotted line separates the regions in which the wide-splitting and the narrow-splitting mode represents the mode with maximal growth rate, respectively.…”

“…This can be done systematically in the vicinity of the codimension-two point (q (ct) , µ (ct) ) where both σ l and σ t are zero to cubic order (cf. (27,29)). There the orientation of the eigenmodes depends solely on the fifth-order terms.…”

“…In these theoretical analyses two types of long-wave instabilities have been identified, a longitudinal and a transverse mode, with the longitudinal mode being relevant only for Rayleigh numbers very close to the saddle-node bifurcation at which the hexagons first appear. In a more detailed analysis also perturbations with arbitrary wavenumber and simulations of the nonlinear evolution ensuing from the instabilities have been included [29]. The instabilities typically lead to the formation of penta-hepta defects [30,31].…”

Penta-Hepta Defect Motion in Hexagonal Patterns

“…Spatial gradients are calculated with respect to slow variable R, and asterisks denote complex conjugate. The applicability of these equations to description of real hexagonal patterns has been discussed earlier 5,11,13].…”