The analysis considers a cold-region brackish lake that develops an ice cover on its surface in winter. As the ice cover thickens due to freezing, black ice grows on the bottom side. The freezing process excludes the salt from the ice, resulting in an increase in dissolved salt concentration in the lake water just below the bottom of the ice. The density profile in this type of lake is then governed by two opposing factors: the water temperature distribution generally produces a stable stratification, whereas the excess salinity produced by exclusion tends to increase the density of the upper layers of the lake, resulting in an unstable contribution to stratification. Competition between these two factors can lead to unstable density gradients, so an initially motionless lake with a slowly thickening plane ice cover develops a convective flow field. This flow field can then feed back into the evolution of the morphology of the ice cover itself, resulting in a morphodynamic interaction between water and ice. This paper aims at investigating this morphodynamic instability. A mathematical model for the stability of a lake covered with an initially plane ice cover growing in time is proposed here. The results reveal threshold values of the main dimensionless parameters, expressed in the form of Rayleigh numbers, and in particular define a ratio between buoyancy effects and diffusivity that governs neutral stability conditions. When the system is unstable, i.e., for Rayleigh numbers above ~103 depending on the values of the input parameters, the analysis predicts the growth of convective flow circulation cells responsible for the morphodynamic evolution of the ice-water interface.