The main purpose of this study is to determine bifurcation as the primary instability of a glass melting furnace. Steady-state and unsteady characteristics of natural convection in the partially open cavity as appeared in a glass melting furnace is investigated by using numerical analysis. Three types of convection, such as steady laminar, unsteady periodic or unsteady quasi-periodic convection may occur according to the temperature difference between upper two isothermal surfaces along the depth of cavity in a glass melting furnace. In the temperature difference of 150-900 K between batch and free surface, the larger the temperature difference, the weaker the convection strength and unsteadiness. Since the glass viscosity is increasing exponentially in the lower temperature, the batch freezes the thermofluidic field especially below the surface of it. If the depth of cavity is 0.5 m, the bifurcation to time-dependent natural convection may occur in the range of 60-650 K. If that is 1.0 m, it may occur in the whole range of temperature difference.