the growth rate is maximized at the lowest ZF wavenumber $|q_{x}|=2\upi/L_{x}$. In other words, the box-scale ZF grows fastest, with the growth rate given by $\gamma_{+}\approx2\gamma_{\boldsymbol{k}}$, i.e., twice the growth rate of the primary DW instability.

![The time history of the DW and ZF energies (\[eq:Energies\]) corresponding to figure \[fig:THhistory\]. The primary and secondary instabilities are clearly seen, with the secondary-instability growth rate being twice the primary-instability growth rate. The black dashed line is the ZF energy calculated from (\[eq:Linear\_Secondary\_Ec\]) that corresponds to the critical ZF amplitude $\varphi_{{\rm c}}$ from (\[eq:Linear\_Secondary\_phic\]). It is seen that this energy roughly corresponds to the onset of the fully nonlinear regime. This value is reached by both $E_{\rm DW}$ and $E_{\rm ZF}$ at approximately the same time.[]{data-label="fig:PI_and_SI"}](figures/PI_and_SI){width="1\columnwidth"}

In the following, we show that this exponential growth of the ZF at the box scale is more common than the super-exponential growth, provided that the characteristic amplitude $\varphi_0$ of the initial random noise is small enough. At first, both the primary DW and the sidebands grow exponentially, $$|\varphi_{\boldsymbol{k}}|\sim|\varphi_{\boldsymbol{k}_{\pm}}|\sim|\varphi_{{\rm 0}}|{\rm e}^{\gamma_{\boldsymbol{k}}t},$$ while the ZF amplitude remains at the noise level. Then, DWs grow for some time $t_{{\rm p}}$ before they begin to affect ZFs. Assume that at 