Transition to branching flows in optimal planar convection

We study steady flows that are optimal for heat transfer in a two-dimensional periodic domain. The flows maximize heat transfer under the constraints of incompressibility and a given energy budget (i.e. mean viscous power dissipation). Using an unconstrained optimization approach, we compute optima starting from 30--50 random initializations across several decades of Pe, the energy budget parameter. At Pe between 10$^{4.5}$ and 10$^{4.75}$, convective rolls with U-shaped branching near the walls emerge. They exceed the heat transfer of the simple convective roll optimum at Pe between 10$^{5}$ and 10$^{5.25}$. At larger Pe, multiple layers of branching occur in the optima, and become increasingly elongated, asymmetrical, and heterogeneous. The rate of heat transfer scales as Pe$^{0.575}$, which, in this range of Pe, is very close to the Pe$^{2/3}(log , $Pe)$^{-4/3}$ behavior of self-similar branching solutions proposed earlier. Compared to the simple convective roll, the branching flows have lower maximum speeds and thinner boundary layers, but nearly the same maximum power density.