function $W(x,k_x)$ of the runaway mode (color), the local $U$ (magenta curve), and the runaway trajectory (dashed curve; see (\[eq:GammaTI\_runaway\_traj\])). Note that we have shifted the coordinates as $x\to x+\pi/q_x$ to recenter the ZF minimum at $x=0$. (b) The structure of each term in (\[eq:GammaTI\_xintegration\]) calculated from $W$ of the runaway mode in (a). (c) The Wigner function $W(x,k_x)$ of the trapped mode (color), the local $U$ (magenta curve), and isosurfaces of $\mathcal{H}$ (dashed contours; see (\[eq:GammaTI\_Hamiltonian\])). In this figure, $\Delta x$ and $\Delta k_x$ denote the characteristic widths of the mode in the $x$ and $k_x$ directions, correspondingly. (d) Same as (b) but for the trapped mode. \[fig:TIstructure\]](figures/modeStructure){width="1\columnwidth"}

![(a) The empirical factor $\eta_{\rm R}$ (\[eq:GammaTI\_runaway\_eta\]) as a function for $\delta_0$: numerical values of a sinusoidal ZF (\[eq:Linear\_Tertiary\_sinusoidalU\]) (blue circles) versus the fitting formula (black curve). The parameters are $\beta=6$, $q_x=0.4$, $u=10$, and $\hat{D}$ given by (\[eq:Equations\_D\]). It is found that $\eta_{\rm R}$ is not sensitive to $u$. (b) Same as (a) except for $\eta_{\rm T}$ (\[eq:GammaTI\_trapped\_eta\]). It is found that $\eta_{\rm T}$ is not sensitive to $u$ at $u< \beta/q_x^2$. (c) The TI growth rates versus $|\mathcal{C}|=q_x^{2}u$ at $\delta_{0}=1.5$, $\beta=6$, $q_{x}=0.4$, and varying $u$. Black curves: numerical solutions of 