  Penetrative turbulence, which occurs in a convectively unstable fluid layer
and penetrates into an adjacent, originally stably stratified layer, is
numerically and theoretically analyzed. We chose the most relevant example,
namely thermally driven flow of water with a temperature around $T_m\approx
4^\circ\rm{C}$, where it has its density maximum. We pick the Rayleigh-B\'enard
geometry with the bottom plate temperature $T_b > 4^\circ\rm{C}$ and the top
plate temperature $T_t \le 4^\circ\rm{C}$. Next to the overall thermal driving
strength set by the temperature difference $\Delta = T_b - T_t$ (the Rayleigh
number $Ra$ in dimensionless form), the crucial new control parameter as
compared to standard Rayleigh-B\'enard convection is the density inversion
parameter $\theta_m \equiv (T_m - T_t ) / \Delta$. The crucial response
parameters are the relative mean mid-height temperature $\theta_c$ and the
overall heat transfer (i.e., the Nusselt number $Nu$). We theoretically derive
the universal (i.e., $Ra$-independent) dependence $\theta_c (\theta_m)
=(1+\theta_m^2)/2$, which holds for $\theta_m$ below a $Ra$-dependent critical
value, beyond which $\theta_c (\theta_m)$ sharply decreases and drops down to
$\theta_c=1/2$ at $\theta_m=\theta_{m,c}$. Our direct numerical simulations
with $Ra$ up to $10^{10}$ are consistent with these results. The critical
density inversion parameter $\theta_{m,c}$ can be precisely predicted by a
linear stability analysis. The heat flux $Nu(\theta_m)$ monotonically decreases
with increasing $\theta_m$ and we can theoretically derive a universal relation
for the relative heat flux $Nu(\theta_m)/Nu(0)$. Finally, we numerically
identify and discuss rare transitions between different turbulent flow states
for large $\theta_m$.
