the eigenmode structures, we also show in figure \[fig:TI\_eigenvalue\] their corresponding Wigner functions $W(x,k_x)$ (\[eq:appA\_Wigner\]) and contour plots of the drifton Hamiltonian $\mathcal{H}$ (\[eq:appB\_Drifton\_Hamiltonian\]). The Wigner function can be understood as the distribution function of “driftons” (DW quanta) in the $(x,k_{x})$ phase space [@Smolyakov99; @Ruiz16; @Zhu18c], and its shape is expected to align with the contours of $\mathcal{H}$ . Then, eigenmodes are naturally centered at phase-space equilibria of $\mathcal{H}$, namely, $$\partial_{x}\mathcal{H}=\partial_{k_{x}}\mathcal{H}=0\quad\Rightarrow\quad U' = k_x=0.\label{eq:Linear_Tertiary_localize}$$ This explains eigenmode localization near extrema of $U$. \[Strictly speaking, (\[eq:Linear\_Tertiary\_localize\]) stems from our approximation of sinusoidal flow (\[eq:Linear\_Tertiary\_sinusoidalU\]), which ensures that $U'$ and $U'''$ become zero at same locations. Nevertheless, (\[eq:Linear\_Tertiary\_localize\]) remains a good approximation as long as ZFs are large-scale, [[i.e., ]{}]{}$|U'''/\bar{k}^2|\ll U'$.\] Maxima of $U$ (even $n$) correspond to phase-space islands encircled by “trapped” trajectories, and minima of $U$ (odd $n$) correspond to saddle points passed by the “runaway” trajectories [@Zhu18a; @Zhu18b; @Zhu18c]. Hence, we call the modes localized near maxima and minima of $U$ trapped and runaway modes, respectively. (See Appendix \[appB\] for more discussions on drifton phase-space trajectories.) In the next section, we provide analytic calculation of the TI growth rates based on the above observations.

Tertiary-instability growth rate {#sec:TIcalculation}
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