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 with a given flow intensity
budget. In the energy-constrained case, where kinetic energy is prescribed,
optimal designs are known to be convection rolls in the large energy limit.
In the enstrophy-constrained case, however, previous numerical studies
indicate a much more complicated flow structure is favorable, and show the
emergence of near-wall recirculation zones beyond a certain critical
enstrophy level. After a brief introduction to the wall-to-wall optimal
transport problem, we describe a useful reformulation inspired by related
questions in homogenization. This leads to a perhaps unexpected connection
between the wall-to-wall problem and questions arising originally in the
study of energy-driven pattern formation in materials science. 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 logarithmic corrections. We discuss implications for the problem of
finding the best absolute upper limits on the rate of heat transport in
turbulent Rayleigh-Bénard convection. This is joint work with C. Doering.