Top-left: height-time profiles of the apex of the envelope fields (upper lines) and the center of the erupting ropes (lower lines): α = 0.3 (dashed line) and α = 0.4 (solid line). Bottom-left: the rising velocity, V_z, of the center of the rope: α = 0.3 (dashed line) and α = 0.4 (dotted line). Top-right: the spatial distribution of the vertical component of the forces for α = 0.3 and t = 43.3   min: magnetic pressure force (denoted by P_m), tension force (denoted by T), force (dashed line) and total force (dotted-dashed line). Bottom-right: as above, for α = 0.4 and t = 48.3   min. The vertical dashed line shows the location of the axis of the rope.