For the trapped mode (labeled with superscript “T”), which corresponds to $\mathcal{C}<0$, one has $$\gamma_{\rm TI}^{\rm T}=\bar{\gamma}+\sqrt{\frac{1}{2}\left(\left|\frac{\beta}{\mathcal{C}}\right|-1\right)}\,\frac{\delta_0k_y^2\mathcal{C}}{(1+k_y^2)^2+\delta_0^2k_y^2}.\label{eq:Analytic_gammaT}$$ Figure \[fig:TIgrowthRate\](c) shows that these formulas are in good agreement with our numerical calculations of the eigenvalues. Also, we have verified that (not shown) the results in figure \[fig:TIgrowthRate\](c) are insensitive to $q_x$ as long as $q_x$ is small, more specifically, $q_x^2\ll 1+k_y^2$.

Notably, while the trapped-mode growth rate always decreases with $|\mathcal{C}|$, the runaway-mode growth rate can increase at large $\mathcal{C}$ if $\delta_0$ is large. In fact, at $\mathcal{C}\gg\beta$, (\[eq:Analytic\_gammaR\]) becomes $$\gamma_{\rm TI}^{\rm R}\approx \left[\frac{\delta_0k_y^2}{(1+k_y^2)^2+\delta_0^2k_y^2}-\sqrt{\frac{1}{2}}\,\frac{(1+k_y^2)k_y}{(1+k_y^2)^2+\delta_0^2k_y^2}\right]\mathcal{C},$$ which predicts that $\gamma_{\rm TI}^{\rm R}$ increases with $\mathcal{C}$ if $\delta_0>(k_y + k_y^{-1})/\sqrt{2}$. This is verified by our numerical calculations for $k_y=1$ (not shown), which predict a similar condition, $\delta_0\gtrsim 1.7$. Therefore, it is possible that the TI can develop in strong ZFs, but the physical mechanism is very different from the KH mode, as will be discussed in section \[subsec:KHI\].

Alternative approach {#subsec:Analytic_WME}
--------------------

![The structure of the tertiary modes with $m=0$ in the zonal-velocity profile (\[eq:Linear\_Tertiary\_sinusoidalU\]). The parameters are $\delta_{0}=1.6$, $\beta=5$, $q_{x}=0.2$, $u=50$ (hence, $\mathcal{C}=\pm 2$), and $\hat{D}$ given by (\[eq:Equations\_D\]). These parameters result in $\gamma_{{\rm TI}}^{{\rm R}}=-0.276$ and $\gamma_{{\rm TI}}^{{\rm T}}=-0.587$. (a) The Wigner 