A weakly nonlinear analysis is employed to investigate the onset of plume convection in the mushy layer of a binary solution directionally solidified from below. An improved mathematical model including the constant pressure condition at the melt/mush interface is applied to analytically analyse the nonlinear behaviour of the convection. Results show that, due to the consideration of the constant pressure condition at the interface, the present analytical results are much better in comparison with the experiments than those obtained by previous studies, in which the no-vertical-flow condition at the interface was considered. It is also shown that the bifurcation to three-dimensional hexagon convection, corresponding to the onset of plume convection, is subcritical. For the case of small concentration ratio [Cscr ] (equation (5b)) the channel of the plume is generated at the top of the mush and grows downwards into the mush. For the case of large [Cscr ], the channel may be generated at the interior of the mush and grow upwards to the top of the mush. The possible parameter regime in which the flow is of a stable down-centre hexagon is discussed.