Fourier amplitudes of the vertical velocities of neutrals (blue dashed line) and charges (red dotted line) at the end of the linear phase at time ≈190.4 s, and the vertical eigenfunction of the velocity obtained in the incompressible MHD approximation (black solid line), as a function of height between −0.2 Mm and 0.2 Mm. Top left panel: L2-S, n = 16; top right panel: L2-α-S, n = 16; bottom left panel: L2-S, n = 32; bottom right panel: L2-α-S, n = 32. The eigenfunctions are normalized to the corresponding maximum of the velocity amplitudes. The profile of the semi-analytical eigenfunction is less smooth as it was obtained on a coarser grid of 1024 points, compared to simulations that used 2048 points in the vertical direction.