The influence of non-resonant modes

The aim of this section is twofold. First, we want to disentangle the different contributions to the fifth order potential and study how they lead to the observed planform dependence of the potential. Second, we want to discuss how fifth order corrections depend on the model parameters g and σ. This will lead to a proposed rescaled model in which the distance to threshold becomes parameter dependent such that fifth order corrections can be neglected throughout the phase diagram.

We start with investigating the influence of the model parametergon fifth order corrections of the potential which are of the formP

i,j,k|A_i|²|A_j|²|A_k|²g_ijk. Stationary amplitudes at third order scale as A ≈ p

r/(1 +g(n−1)) and fifth order corrections in general increase these

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Figure 6.9.: Phase diagram with the inclusion of higher order corrections. Shown are the regions where a certain planform has the minimal energy Eq. (6.37). (a)r = 0.001, (b) r= 0.01 (c)r = 0.1. Colored regions: LDP has minimal energy, white region: stripes have minimal energy, gray region: potential is not bounded from below.

amplitudes. In the following we assume A_j− ≈0 and small amplitude modulations A_j ≈ A. The sum over all fifth order coupling coefficients scales for largeσasS =P

i,j,kg_ijk∝(g−1)². Therefore fifth order contributions increase at least quadratically with g, and in the limit g→1 these corrections vanish. The relative energy change in the potential due to fifth order corrections is shown in Fig. 6.10(a) for different values of g. We observe that for g close to one, the potential is clustered according to the number of flipped modes clusters κ. This clustering results from the clustering of the quantity S, see Fig. 6.10(c). For smaller values of g, however, we observe deviations of the clustering which result from modulations in the stationary amplitudes.

With an increasing number of active modes or with an increasingσthe influence of fifth order corrections increase. How can an increase in n be captured by moving closer to threshold i.e. reducing r? To answer this question we consider the distribution of modes off the critical circle which contribute to the fifth order coupling coefficients. The coupling terms in Eq. (6.19), see also Appendix A.5, are composed of contributions from the nonlinearity and from the inverse linear operator ( ˆL⁰)⁻¹ acting on the modes off the critical circle. We want to reveal the influence of these modes on the energy levels for different ECP solutions. The configuration non-resonant modes for three n = 5 ECP solutions is illustrated in Fig. 6.11.

0 1 1 1 1 1 2 2 2 2 2 3

Figure 6.10.: Lifting of planform degeneracy for all n=8 planforms. (a) Relative energy change due to fifth order corrections as a function of the number of flipped modes, (b) as a function of the planform anisotropy. Red: g= 0.98, σ= 1.4 Λ, blue:

Their configuration and thus their contribution to the potential is planform dependent. For instance, in the case of the LDP solution the modes come closer to the critical circle, as for the fully symmetric HDP solution. At fifth order there aren³−2n²+nmodes off the critical circle but not all contribute equally. Due to the shape of ( ˆL⁰)⁻¹, the closer the modes off the critical circle come to the critical circle, the more they influence the couplings in the amplitude equations. To study the influence of these modes on the coupling functiong_ijk, we mainly concentrate on those modes close to the critical circle which contribute most. We sort the modes according to their distance to the critical circle~k⁽ⁱ⁾s , i= 1,2, ..., where the modes

~k⁽¹⁾s are closest. Note, that the largest contributions to the amplitude equations result not

Figure 6.11.: Distribution of modes off the critical circle contributing to the n= 5 amplitude equations at fifth order, kc= 1. (a)l= (1,−1,1,−1,1) (b)l= (1,1,1,−1,1) (c) l= (1,1,1,1,1). Active modes are marked in red. Near resonant modesks⁽¹⁾ and k⁽²⁾s are marked in green and orange, respectively.

from the modes closest to the critical circle, but from modes with|k_s|< k_c. We further define

L_i = X

where the sum is performed over all modesk⁽ⁱ⁾_s with a fixed distance to the critical circle. We calculate the contribution to the coupling function and thus to the potential that results from these near resonant modes. For alln= 8 planform solutions we plotLi where we successively included more and more modes off the critical circle, starting with the closest ones. This is shown in Fig. 6.10(d). We observe that the clustering according to κ observed in the fifth order potential is already present in the contributions from the near resonant modes.

Modes that have the largest coefficient from ( ˆL⁰)⁻¹ are given byk⁽¹⁾_s =k_i+k_i−2−k_i−1 with

These modes are marked in orange in Fig. 6.11. L1 grows with nas n⁴ and their number is given by nfor a LDP. Therefore L₁ grows withn asn⁵, see orange line in Fig. 6.12(b). The second largest contribution to ( ˆL⁰)⁻¹ results from modes with the minimal distance to the critical circlek⁽²⁾s = 2k_i−k_i−1 with

These modes are marked in green in Fig. 6.11. L2 also grows with nasn⁴ and their number is given by 2(n−1)−4κ. Thus L2 grows as n⁵, see green line in Fig. 6.12(b). The length of the modes k⁽¹⁾_s and k_s⁽²⁾ with increasing number of active modesn is shown in Fig. 6.12(a).

Their contribution to the amplitude equations in terms of L_i, see Eq. (6.38), is shown in

4 6 8 10 12 14 16 18 20 (green). (b) L₁ (orange) and L₂ (green), see Eq. (6.38). Contribution of all off circle modes P

iL_i (blue). Potential at third ((−V₃)^(1/5), black) and fifth ((−V)^(1/5), red) order, parameters: g= 0.9, n= 2πσ/Λ.

Fig. 6.12(b). Finally, the most distant modes are given by k_s^(d) = 2k₁−k_n−1 with |k_s^(d)|= 3 and Ld = −₆₄¹. We confirmed our analysis by calculating the third (black line) and fifth order (red line) potential with an increasing number of active modes, see Fig. 6.12(b). Indeed the potential at fifth order increases with nas n⁵. Note, when nis a multiple of three triad resonance occur and the potential receives additional contributions. To compensate the fifth order corrections we propose a rescaled model where the bifurcation parameter r has to be model dependent i.e. r =r(σ/Λ, g). The dependence of the amplitudesA on the parameter σ is negligible if the solution is not close to its stability borders. As A ∝ √

r+r^3/2 and the stability borders scale for large n as n ≈ 2πσ/Λ, the bifurcation parameter for a given g should scale as (√

r+r^3/2)⁶(σ/Λ)⁵ = const. Furthermore, we revealed the dependence of the potential on the number of flipped mode clustersκ. Contributions from the inverse linear operator acting on modes off the critical circle are clustered according to κ, see Fig. 6.10(d).

If we take the contribution of the nonlinearity into account, see Fig. 6.10(c), this clustering is preserved. The potential finally is clustered according to κ only for g close to 1. For smallerg, modulations in the stationary amplitudes lift the approximate degeneracy inκ, see Fig. 6.10(a).