\qquad \qquad \qquad
\qquad \qquad \qquad \ \ \,   \label{eq:els3}$$ where $P = p + \mathbf{b}_{\perp}^2/2 $ is the total pressure, and is linked to the nonlinear terms by incompressibility (\[eq:els3\]): $$\label{eq:els4}
\mathbf{\nabla}_{\perp}^2 P =
- \sum_{i,j=1}^2 \Big( \partial_i z_j^- \Big) \Big( \partial_j z_i^+ \Big).$$

In terms of the Elsässer variables $\mathbf{z}^{\pm} = \mathbf{u}_{\perp} \pm \mathbf{b}_{\perp}$, a velocity pattern $\mathbf{u}_{\perp}^{0,L}$ at upper or lower boundary surface becomes the constraint $\mathbf{z}^{+} + \mathbf{z}^{-} = 2 \mathbf{u}_{\perp}^{0,L}$ at that boundary. Since, in terms of characteristics (which in this case are simply $\mathbf{z}^{\pm}$ themselves), we can specify only the incoming wave (while the outgoing wave is determined by the dynamics inside the computational box), this velocity pattern implies a reflecting condition at the top ($z=L$) and bottom ($z=0$) planes: $$\label{eq:bc0}
\mathbf{z^{-}}  = - \mathbf{z^{+}} + 2 \, \mathbf{u}^{0}_{\perp}
\quad \textrm{at} \ z=0,$$ $$\label{eq:bcL}
\mathbf{z^{+}} = - \mathbf{z^{-}} + 2 \, \mathbf{u}^{L}_{\perp}
\quad \textrm{at} \ z=L.$$

The linear terms ($\propto \partial_z$) in equations (\[eq:els1\])-(\[eq:els2\]) give rise to two distinct wave equations for the $\mathbf{z}^{\pm}$ fields, which describe Alfvén waves propagating along the axial direction $z$. This wave propagation, which is present during both the linear and nonlinear regimes, is responsible for the continuous energy 