the equations of reduced MHD (RMHD) [@kp74; @stra76; @mont82].

These equations are valid for a plasma with small ratio of kinetic to magnetic pressures, in the limit of a large loop-aspect ratio ($\epsilon = l/L \ll 1$, $L$ being the length of the loop and $l$ being the minor radius of the loop) and of a small ratio of poloidal to axial magnetic field ($b_\perp /B_0 \le \epsilon$). In dimensionless form they can be written as: $$\begin{gathered}
\frac{\partial \boldsymbol{u}_{\perp}}{\partial t} 
+ \left(  \boldsymbol{u}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right)  \boldsymbol{u}_{\perp}
= - \boldsymbol{\nabla}_{\perp} \left( p + \frac{\boldsymbol{b}_{\perp}^2}{2} \right) \\
+   \left(  \boldsymbol{b}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{b_{\perp}} 
+  c_{\mathcal{A}} \, \frac{\partial \boldsymbol{b}_{\perp}}{\partial z}  
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp} \boldsymbol{u}_{\perp}, \label{eq:adim1}\end{gathered}$$ $$\begin{gathered}
\frac{\partial \boldsymbol{b}_{\perp}}{\partial t} =  \left(  \boldsymbol{b}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{u_{\perp}} -  \left(  \boldsymbol{u}_{\perp} \cdot \boldsymbol{\nabla}_{\perp} \right) \boldsymbol{b _{\perp}} \\
+  c_{\mathcal{A}} \,  \frac{\partial \boldsymbol{u}_{\perp}}{\partial z} 
+ \frac{(-1)^{n+1}}{Re_n} \boldsymbol{\nabla}^{2n}_{\perp} 
\boldsymbol{b}_{\perp}, \label{eq:adim2}\end{gathered}$$ $$\boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{u}_{\perp} = 0, \qquad
\boldsymbol{\nabla}_{\perp} \cdot \boldsymbol{b}_{\perp} = 0, \qquad \qquad \qquad \quad
\label{eq:adim3}$$ where $\mathbf{u}_{\perp}$ and $\mathbf{b}_{\perp}$ are the components of the velocity and magnetic fields perpendicular to the mean field, and $p$ is the kinetic pressure. The gradient operator likewise has only components in the $x$-$y$ plane perpendicular to the axial 