  Three-dimensional convection driven by internal heat sources and sinks (CISS)
leads to experimental and numerical scaling-laws compatible with a
mixing-length - or `ultimate' - scaling regime $Nu \sim \sqrt{Ra}$. However,
asymptotic analytic solutions and idealized 2D simulations have shown that
laminar flow solutions can transport heat even more efficiently, with $Nu \sim
Ra$. The turbulent nature of the flow thus has a profound impact on its
transport properties. In the present contribution we give this statement a
precise mathematical sense. We show that the Nusselt number maximized over all
solutions is bounded from above by const.$\times Ra$, before restricting
attention to 'fully turbulent branches of solutions', defined as families of
solutions characterized by a finite nonzero limit of the dissipation
coefficient at large driving amplitude. Maximization of $Nu$ over such branches
of solutions yields the better upper-bound $Nu \lesssim \sqrt{Ra}$. We then
provide 3D numerical and experimental data of CISS compatible with a finite
limiting value of the dissipation coefficient at large driving amplitude. It
thus seems that CISS achieves the maximal heat transport scaling over fully
turbulent solutions.
