us examine the validity of our approximation in (\[eq:AnalyticTI\_oscillator\_Hexpand\]). First, the parabolic approximation of $U$ is valid if the mode spatial width in $x$, which is determined by $\mathcal{C}$, is much smaller than $q_x^{-1}$, which is the characteristic scale of ZFs. Specifically, for the sinusoidal ZF (\[eq:Linear\_Tertiary\_sinusoidalU\]), we have $\mathcal{C}=q_x^2 u$, so the parabolic approximation (\[eq:AnalyticTI\_oscillator\_Uexpand\]) is valid at small enough $q_x$ and large enough $u=\mathcal{C}/q_x^2$. Second, the expansion of $\hat{k}^{-2}$ in (\[eq:AnalyticTI\_oscillator\_Kexpand\]) is valid at $|k_x^2|\ll |k_0^2|$, where $k_x$ is the characteristic mode wavenumber in $x$. From (\[eq:AnalyticTI\_oscillator\]), $k_x$ can be estimated as $k_x=1/\sqrt{\tau}$. Then, the requirement $|k_x^2|\ll |k_0^2|$ leads to $|\sqrt{2(1+\beta/\mathcal{C})}|\gg 1$, or equivalently, $|\mathcal{C}|\ll\beta$, and one expects that the approximation in (\[eq:AnalyticTI\_oscillator\_Kexpand\]) becomes invalid as $|\mathcal{C}|$ approaches $\beta$. Therefore, in the following, we restrict our consideration to the parameter regime $|\mathcal{C}|<\beta$, which is also the regime relevant to our numerical simulations.

The TI growth rate $\gamma_{\rm TI}$ is obtained by taking the imaginary part of $\omega$. Within the regime $|\mathcal{C}|<\beta$, let us introduce the notation $$\bar{\gamma}\doteq{\rm Im}\,\bar{\Omega}=\frac{\delta_0k_y^2(\beta+\mathcal{C})}{(1+k_y^2)^2+\delta_0^2k_y^2}-D_0,$$ which is the primary-instability growth rate $\gamma_{\boldsymbol{k}}$ (\[eq:Linear\_Primary\_Omegak\]) modified by $\mathcal{C}$. Then, for the runaway mode (labeled with superscript “R”), which corresponds to $\mathcal{C}>0$, one has $$\gamma_{\rm TI}^{\rm R}=\bar{\gamma}-\sqrt{\frac{1}{2}\left(1+\frac{\beta}{\mathcal{C}}\right)}\,\frac{(1+k_y^2)k_y\mathcal{C}}{(1+k_y^2)^2+\delta_0^2k_y^2}. \label{eq:Analytic_gammaR}$$ 