This article presents a mathematical model of the plane evolution of alluvial meandering streams, through downstream migration and lateral expansion of meander loops. Under the conditions prevailing in natural streams, the channel centerlines follow sine-generated curves, with an assumed steady-state turbulent and subcritical flow, of large width-to-depth ratio (≥15, for example) and small Froude number (Fr). The plane deformation of the channel is caused by the action on the banks of the convective vertically-averaged meandering flow. The growth (migration and expansion) of meander loops is attributed to the regime-trend. The computational results of the model show that the obtained migration and expansion patts of the meander loops are in good agreement with those of observations and measurements in similar meandering streams.