Identify the mechanism governing the anomalous Fourier mode

Identify the mechanism responsible for the appearance and changing dynamical role of the anomalous Fourier mode \(\tilde\zeta_{2,-1}\) in the two-frequency-driven Faraday-wave simulations, including why it is transient in some SSS-I transitions, absent in others, and persistent in the lower-forcing-amplitude modulated regime.

Background

The anomalous mode $\tilde\zeta_{2,-1}$, also denoted earlier by the equivalent index convention $\tilde\zeta_{-2,1}$, has wavenumber $\sqrt{13}k_c/4$ and is not part of a hexagonal triad on the rectangular computational grid. The simulations show that it can become prominent during the transition toward SSS-I, but its behavior is not consistent across runs or forcing amplitudes.

In the concluding discussion, the authors note that this mode is transient in one simulation, absent in another, and permanent in the modulated state obtained at lower forcing amplitude. They explicitly state that its role is unresolved.