d}/{\rm d}x,\quad\hat{\bar{k}}^{2}=1+k_{y}^{2}+\hat{k}_x^2-{\rm i}\delta_{0}k_{y}.\label{eq:Linear_Tertiary_khat}$$ If an eigenvalue $\omega$ exists and $$\gamma_{{\rm TI}}\doteq\Imag\,\omega>0,$$ then the perturbation grows exponentially. This is the TI.

Equation (\[eq:Linear\_Tertiary\_linearDW\]) does not have an analytic solution for an arbitrary profile $U$, but a general understanding can be developed by considering special cases. In [@Zhu18c], we considered the ZF velocity profile $$U(x)=u\cos q_{x}x,\label{eq:Linear_Tertiary_sinusoidalU}$$ with $\hat{\delta} = \hat{D} = 0$. In this case, the system exhibits an instability of the KH type provided that $q_{x}^{2}>1$ and $q_x^2u>\beta$. In [@Zhu18c], we also discussed a generalization to periodic nonsinusoidal profiles. However, generalizing those results to nonzero $\hat{\delta}$ and $\hat{D}$ is challenging. The common approach is to adopt the 4MT again, [[i.e., ]{}]{}to assume a DW perturbation with $\boldsymbol{k}=(0,k_{y})$ and two sidebands with $\boldsymbol{k}_{\pm}=(\pm q_{x},k_{y})$ as small perturbations [@Kim02; @St-Onge17; @Rath18; @Zhu18a]. In particular, [@St-Onge17] derived $\gamma_{{\rm TI}}$ within the 4MT and estimated the Dimits shift by finding a sufficient condition for $\gamma_{{\rm TI}}=0$. However, the 4MT-based approach is not entirely satisfactory, because the ZF is typically far from sinusoidal, as seen in simulations. Even more importantly, the 4MT approach ignores the fact that there are multiple TI modes with different growth rates. As we show below, understanding the variety of these modes 