The differences between the interdecadal variability under mixed and constant flux boundary condition are investigated using a coarse-resolution ocean model in an idealized flat-bottom single-hemisphere basin. We determine objective features that allow to distinguish one type of oscillation versus the other. First, by performing a linear stability analysis of the steady state obtained under restoring boundary conditions, we show that the interdecadal variability under constant flux and mixed boundary conditions arises respectively from the instability of a linear mode around the mean stratification and circulation, and from departure from initial state. Based on the budgets of density variance, we show next that the two types of oscillations have different enery sources: Under constant-flux boundary condition (the thermal mode), the downgradient meridional eddy heat flux in the western boundary current regions sustains the interdecadal variability, whereas under mixed boundary condition (the salinity mode), a positive feedback between convective adjustment and restoring surface heat flux is at the heart of the existence of the decadal oscillation. Furthermore, the positive correlations between temperature and salinity anomalies in the forcing layer are shown to dominate the forcing of density variance. In addition, the vertical structure of perturbations reveals vertical phase lags at different depths in all tracer fields under constant flux, while under mixed boundary conditions, only the temperature anomalies show a strong dipolar structure. We propose that these differences will allow to identify which type of oscillation, if any, is at play in the more exhaustive climate models.