simple form: $${\rm i}\hat{\delta}\doteq{\rm i}\delta_{0}\hat{k}_{y}\equiv\delta_{0}\partial_{y},\label{eq:Equations_delta}$$ with $\delta_{0}$ being a positive constant. (This can be used to model trapped-electron dynamics [@Tang78].) Finally, the operator $\hat{D}$ models damping effects such as viscosity. Following [@St-Onge17], we use $$\hat{D}=1-0.01\nabla^{2}\label{eq:Equations_D}$$ throughout this paper. (An exception is made in section \[sec:DSPrediction\], where another form of $\hat{D}$ is introduced for comparison.) Here, the first (friction) term is added in order to prevent possible energy build up at large scale, as is also done by @St-Onge17. (As will be seen from our results below, this term also increases the Dimits shift and thus facilitates its numerical observation.) Note that due to $\hat{\alpha}$ in front of $\hat{D}$ in (\[eq:Equations\_mTHE\]), the damping applies only to DWs, while ZFs are left collisionless. Then, the Dimits regime can be defined unambiguously as the regime where ZFs persist forever and the DW amplitude decreases to zero at $t \to \infty$.

Beyond the Dimits regime, DWs are not suppressed and ZFs always keep evolving in the mTHE model, as demonstrated by [@St-Onge17]. To understand the ZF dynamics, we take the zonal average of (\[eq:Equations\_mTHE\]) and obtain $$\partial_{t}U=-\partial_{x}\langle\tilde{v}_{x}\tilde{v}_{y}\rangle-\langle\tilde{v}_{x}{\rm i}\hat{\delta}\tilde{\varphi}\rangle+\mathcal{T}(t),\quad U(x,t)\doteq\partial_{x}\langle\varphi\rangle.\label{eq:Equations_Ut}$$ Here, $U$ is the ZF velocity along $y$, $(\tilde{v}_{x},\tilde{v}_{y})\doteq(-\partial_{y}\tilde{\varphi},\partial_{x}\tilde{\varphi})$ is the $\boldsymbol{E}\times\boldsymbol{B}$ velocity of DW 