equation that describes DW turbulence in slab geometry with coordinates $\boldsymbol{x}=(x,y)$, where $x$ is the radial coordinate and $y$ is the poloidal coordinate: $$\label{eq:Equations_mTHE}
\partial_{t}w+\{\varphi,w\}-\beta\partial_{y}\varphi+\hat{\alpha}\hat{D}w=0,$$ where $$w=\nabla^{2}\varphi-n,\quad n=(\hat{\alpha}-{\rm i}\hat{\delta})\varphi.$$ Here, the system is assumed to be immersed in a uniform magnetic field perpendicular to the $(x,y)$ plane. The ions are assumed cold while the electrons are assumed to have a finite temperature $T_{{\rm e}}$. The plasma has an equilibrium density profile $n_{0}(x)$, which is parameterized by the positive constant $\beta\doteq a/L_{n}$, where $a$ is a reference length and $L_{n}\doteq(-{\rm d}\ln n_{0}/{\rm d}x)^{-1}$ is the scale length of the density gradient. (We use $\doteq$ to denote definitions.) Time is normalized by $a/c_{{\rm s}}$, where $c_{{\rm s}}\doteq\sqrt{T_{{\rm e}}/m_{{\rm i}}}$ is the ion sound speed. Length is normalized by the ion sound radius $\rho_{{\rm s}}=c_{{\rm s}}/\Omega_{{\rm i}}$, where $\Omega_{{\rm i}}$ is the ion gyro-frequency. The electrostatic potential fluctuation $\varphi$ is normalized by $T_{{\rm e}}\rho_{{\rm s}}/ea$ where $e$ is the unit charge, the electron density fluctuation $n$ is normalized by $n_{0}\rho_{\rm s}/a$, and $w$ can be considered as minus the ion guiding-center density [@Krommes00]. The Poisson bracket is defined as $$\{\varphi,w\}\doteq\boldsymbol{v}\cdot\nabla w,\quad\boldsymbol{v}\doteq\hat{\boldsymbol{z}}\times\nabla\varphi,\label{eq:Poisson}$$ which describes nonlinear advection of $w$ by the $\boldsymbol{E}\times\boldsymbol{B}$ flow with velocity 