is essential for understanding the Dimits shift.

Let us assume the same sinusoidal ZF profile (\[eq:Linear\_Tertiary\_sinusoidalU\]) as in @St-Onge17 for now, and let us calculate the corresponding eigenmodes (\[eq:Linear\_Tertiary\_eigen\]) numerically, assuming periodic boundary conditions $x$. In this case, we can search for solutions in the form $$w(x)=\sum_{n=-N}^{N}w_{n}{\rm e}^{{\rm i}nq_{x}x},$$ where $N$ is some large enough integer. In other words, we truncate the Fourier series by keeping only the first $2N+1$ Fourier modes. This turns (\[eq:Linear\_Tertiary\_eigen\]) into a vector equation for $\{w_{-N}, \dots w_0, \dots w_N\}$, where $\hat{H}$ becomes a $(2N+1)\times(2N+1)$ matrix. Then, one finds $2N+1$ eigenmodes with complex eigenfrequencies. Typical numerical eigenmodes are illustrated in figure \[fig:TI\_eigenvalue\]. It is seen that the TI-mode structure is localized at the maximum ($x=0$) or minimum ($x=-\upi/q_{x}$) of the ZF velocity and has either even or odd parity because of the symmetry of $U$. Within the figure, the eigenmodes localized at the ZF minimum can be labeled by the integer $m=0,1,2,\dots$, which also indicates the parity of $w(x)$. Eigenmodes localized near the ZF maximimum can be labeled similarly. Note that in order for a mode to be localized, the ZF must be large-scale, namely, $q_x^2\ll 1+k_y^2$, which is consistent with numerical simulations.

Apart from 