From the summary: 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 analyze 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.