### Abstract or Additional Information

The wall-to-wall optimal transport problem asks for the design of an incompressible flow between parallel walls that most efficiently transports heat from one wall to the other, subject to a prescribed flow intensity budget. In the energy-constrained case, where the kinetic energy is prescribed, optimal designs are known to be convection rolls in the large energy limit. In the enstrophy-constrained case, numerical studies performed by P. Hassanzadeh, G. Chini, and C. Doering and separately by A. Souza, indicate a much more complicated flow structure is favorable in the large enstrophy limit. In particular, these authors observe the emergence of recirculation zones near the walls whose existence is left unexplained. After a brief introduction, we describe a useful reformulation of the wall-to-wall problem inspired by related questions in homogenization theory. This leads to an unexpected connection between the wall-to-wall problem and questions from the study of energy-driven pattern formation in materials science. We illustrate this connection with a few key examples. The result is a new multiple scales construction for the enstrophy-driven wall-to-wall problem which goes beyond the complexity observed in the numerical studies, and achieves the optimal rate of transport in the large enstrophy limit up to possible logarithmic corrections. We discuss implications for the problem of finding the best absolute upper limits on the rate of heat transport in turbulent Rayleigh-Bernard convection. This is joint work with C. Doering.