$t=t_{{\rm p}}$, the box-scale ZF with the amplitude $\varphi_{\boldsymbol{q}}\sim\varphi_{0}$ starts to grow with the growth rate $\gamma_{+}\approx2\gamma_{\boldsymbol{k}}$; then, $\delta_{+}=2\delta_{0}k_{y}\gg q_{x}^{2}$, and we have from (\[eq:Linear\_Secondary\_4MT-a\]) that $$|\partial_{t}\varphi_{\boldsymbol{q}}|\sim 2\gamma_{\boldsymbol{k}}|\varphi_{{\rm 0}}|\approx\frac{2|\varphi_{\boldsymbol{k}}||\varphi_{\boldsymbol{k}_{+}}|k_{y}\delta_{+}}{q_{x}}\approx\frac{2k_{y}\delta_{+}|\varphi_{{\rm 0}}{\rm e}^{\gamma_{\boldsymbol{k}}t_{{\rm p}}}|^{2}}{q_{x}}.\label{eq:Linear_Secondary_phi0}$$ This leads to $$C,D\propto\frac{q_{x}\gamma_{\boldsymbol{k}}|\varphi_{0}|}{2\delta_{0}k_{y}^{2}}.$$ Therefore, $C$ and $D$ are small when the initial noise level $|\varphi_{0}|$ is small enough; hence, the assumptions made above are self-consistent, namely, $A$ and $B$ are indeed much larger than $C$ and $D$, and the box-scale ZF with wavenumber $q_{x}=2\pi/L_{x}$ grows fastest with the growth rate $2\gamma_{\boldsymbol{k}}$.

The secondary instability will persist for some time $t_{{\rm s}}$ until ZFs grows up to a finite amplitude that is enough to significantly distort the DW structure. Using the result from [@Zhu18b], this amplitude can be estimated as follows (also see (\[eq:appB\_U\_critical\])): $$\varphi_{{\rm c}}=\frac{\beta/q_{x}}{2(1+k_{y}^{2})-q_{x}^{2}}.\label{eq:Linear_Secondary_phic}$$ At $\varphi_{\boldsymbol{q}}\ll\varphi_{{\rm c}}$, DWs do not “see” the ZF and hence keep growing exponentially, while at $\varphi_{\boldsymbol{q}}\gtrsim\varphi_{{\rm c}}$ the system enters the fully nonlinear regime. Therefore, $t_{{\rm s}}$ is the time when the ZF amplitude grows from $\varphi_{0}$ to $\varphi_{{\rm c}}$, and it can be estimated as follows: $$t_{{\rm s}}=\frac{1}{2\gamma_{\boldsymbol{k}}}\ln\frac{\varphi_{{\rm c}}}{\varphi_{0}}.\label{eq:ts}$$ Note that (\[eq:Linear\_Secondary\_phic\]) is obtained from the modified Hasegawa–Mima system, so it is based on the assumption that $\delta_0 = 0$. 