Exact relations between Rayleigh-B\'enard and rotating plane Couette flow in 2D

Relying on an exact relationship between Rayleigh-B\'enard Convection (RBC) and Rotating Plane Couette Flow (RPC) restricted to two spatial independent variables, we deduce several relations between both flows: (i) Heat and angular momentum transport differ by $(1-R_\Omega)$, explaining why angular momentum transport is not symmetric around $R_\Omega=1/2$ even though the relation between $Ra$ and $R_\Omega$ has this symmetry. This relationship leads to a predicted value of $R_\Omega$ that maximizes the angular momentum transport that agrees remarkably well with existing numerical simulations of the full 3D system. (ii) One variable in both flows satisfy a maximum principle i.e., the fields' extrema occur at the walls. Accordingly, backflow events in shear flow \emph{cannot} occur in this quasi two-dimensional setting. (iii) For free slip boundary conditions on the axial and radial velocity components, previous rigorous analysis for RBC implies that the azimuthal momentum transport in RPC is bounded from above by $Re_S^{5/6}$ with a scaling exponent smaller than the anticipated $Re_S^1$.