mTHE (\[eq:Equations\_mTHE\]) with $\delta_0=1.5$ (see (\[eq:Equations\_delta\])) at (a) $\beta=4.5$ (first row) and (b) $\beta=6.5$ (second row). The simulation domain size is $L_{x}=L_{y}=20\upi$, with the corresponding numbers of grid points being $N_x=128$ and $N_y=64$, respectively. Periodic boundary conditions are used in both directions, and the nonlinear term is treated using the pseudospectral method with 2/3 dealiasing rule [@boyd2001chebyshev]. The initial conditions are random noise with a small amplitude. Shown are the fluctuations $\tilde{w}$ (colorbar) and the ZF velocity $U$ (green curve) at three different moments of time. It is seen that at $\beta=4.5$, the DW amplitude decreases down to zero (Dimits regime), while at $\beta=6.5$, fluctuations remain strong and ZFs keep evolving.[]{data-label="fig:THhistory"}](figures/THhistory){width="1\columnwidth"}

Primary instability
-------------------

It is straightforward to show that $\lbrace \varphi, w \rbrace = 0$ for Fourier eigenmodes of the form $$\varphi=\varphi_{\boldsymbol{k}}{\rm e}^{{\rm i}\boldsymbol{k}\cdot\boldsymbol{x}-{\rm i}\Omega_{\boldsymbol{k}}t}+{\rm c.c.},$$ where $\boldsymbol{k}=(k_{x},k_{y})$. Therefore, a Fourier eigenmode is an exact solution of the system provided that $\Omega_{\boldsymbol{k}}$ satisfies the following relation: $$\label{eq:Linear_Primary_dispersion}
\omega_{\boldsymbol{k}}\doteq\Real\,\Omega_{\boldsymbol{k}}=\frac{\beta k_{y}(1+k^{2})}{(1+k^{2})^{2}+\delta_{0}^{2}k_{y}^{2}},\quad
\gamma_{\boldsymbol{k}}\doteq\Imag\,\Omega_{\boldsymbol{k}}=\frac{\beta\delta_{0}k_{y}^{2}}{(1+k^{2})^{2}+\delta_{0}^{2}k_{y}^{2}}-\alpha_{\boldsymbol{k}}D_{\boldsymbol{k}}.\label{eq:Linear_Primary_Omegak}$$ Here, $k^{2}\doteq k_{x}^{2}+k_{y}^{2}$, $D_{\boldsymbol{k}}=1+0.01k^{2}$, and we have used (\[eq:Equations\_delta\]). Also, $\alpha_{\boldsymbol{k}}=1$ for $k_{y}\neq0$ and $\alpha_{\boldsymbol{k}}=0$ for $k_{y}=0$, and hence a ZF ($k_{y}=0$) corresponds to $\Omega_{\boldsymbol{k}}=0$, [[i.e., ]{}]{}to a stationary state. From (\[eq:Linear\_Primary\_Omegak\]), it is seen that when $D_{\boldsymbol{k}}=0$, $\gamma_{\boldsymbol{k}}$ 