  We study the dynamical regimes of a density-stratified fluid confined between
isothermal no-slip top and bottom boundaries (at temperatures $T_t$ and $T_b$)
via direct numerical simulation. The thermal expansion coefficient of the fluid
is temperature dependent and chosen such that the fluid density is maximum at
the inversion temperature $T_b>T_i>T_t$. Thus, the lower layer of the fluid is
convectively unstable while the upper layer is stably stratified. We show that
the characteristics of the convection change significantly depending on the
degree of stratification of the stable layer. For strong stable stratification,
the convection zone coincides with the fraction of the fluid that is
convectively unstable (i.e. where $T>T_i$), and convective motions consist of
rising and sinking plumes of large density anomaly, as is the case in canonical
Rayleigh-Benard convection; internal gravity waves are generated by turbulent
fluctuations in the convective layer and propagate in the upper layer. For weak
stable stratification, we demonstrate that a large fraction of the stable fluid
(i.e. with temperature $T<T_i$) is instead destabilized and entrained by
buoyant plumes emitted from the bottom boundary. The convection thus mixes cold
patches of low density-anomaly fluid with hot upward plumes, and the end result
is that the $T_i$ isotherm sinks within the bottom boundary layer and that the
convection is entrainment-dominated. We provide a phenomenological description
of the transition between the regimes of plume-dominated and
entrainment-dominated convection through analysis of quantities for different
stratification strengths. We find that the effect of the stable layer on the
convection decreases only weakly with increasing stratification strength,
meaning that the dynamics of the stable layer and convection should be studied
self-consistently in a wide range of applications.
