show that the asymptotic behavior of the solution at large $|x|$ is $$w(x)\sim {\rm e}^{S(x)},\quad S(x)=-\frac{x^2}{2\tau}=-\left[\frac{{\rm i}(1+k_y^2)+\delta_0k_y}{2\sqrt{2(1+\beta/\mathcal{C})}}\right]x^2.\label{eq:AnalyticTI_S}$$ To ensure that $w\to 0$ at large $|x|$, we require ${\rm Im}\sqrt{1+\beta/\mathcal{C}}\,>0$ if $1+\beta/\mathcal{C}<0$. We also assumed that $\delta_0,k_y>0$. Then, letting $w=\phi(x)\exp S(x)$, we obtain $$\phi''-\frac{2x}{\tau}\phi'+\frac{\varepsilon-\tau}{\tau^2}\phi=0.\label{eq:Hermite}$$ Solutions are $\phi=\mathsf{H}_m(x/\sqrt{\tau})$, where $\mathsf{H}_m$ are Hermite polynomials, $m=0,1,2,\dots$, and $$\varepsilon=(2m+1)\tau.$$ Therefore, for each sign of $\mathcal{C}$, eigenmodes are labeled by $m$. In figure \[fig:TI\_eigenvalue\], these approximate solutions are compared with numerical solutions of (\[eq:Linear\_Tertiary\_eigen\]). In the following, we shall focus on the two modes with $m = 0$, since they are most unstable. In this case, $\phi=\mathsf{H}_0$ is constant and $\varepsilon=\tau$. This corresponds to $\tilde{w}=\Real[\exp(S+{\rm i}k_y y)]$, and the eigenfrequencies are found from (\[eq:AnalyticTI\_oscillator\_coefficients\]) to be $$\omega=\bar{\Omega}+k_yU_0-\frac{{\rm i}k_y\mathcal{C}\sqrt{(1+\beta/\mathcal{C})/2}}{1+k_y^2-{\rm i}\delta_0 k_y},\quad \bar{\Omega}=\frac{k_y(\beta+\mathcal{C})}{k_0^2}-{\rm i}D_0.\label{eq:AnalyticTI_oscillator_omega}$$ Here, $\bar{\Omega}$ is the primary-mode eigenfrequency $\Omega_{\boldsymbol{k}}$ (\[eq:Linear\_Primary\_Omegak\]) modified by $\mathcal{C}$, $k_yU_0$ is the local Doppler shift, and the remaining term in $\omega$ vanishes at zero $\mathcal{C}$. Note that at $\mathcal{C}=0$, $\omega$ reduces to the primary-mode frequency $\Omega_{\boldsymbol{k}}$ at $\boldsymbol{k}=(0,k_y)$. Hence, TI modes found here can be interpreted as standing primary modes modified by ZFs. Accordingly, the TI growth rate $\gamma_{\rm TI}$ approaches the primary-instability growth rate in the limit $\mathcal{C} \to 0$.

Let 