Improved bounds on horizontal convection

For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by C Ra-1/3 as the horizontal convective Rayleigh number Ra -> infinity for some constant C (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55-70). We re-examine the variational arguments leading to this `ultimate regime' by using the Wentzel-Kramers-Brillouin method to solve the variational problem in the Ra -> infinity limit and exhibiting solutions that achieve the ultimate Ra(1/)3 scaling. As expected, the optimizing flows have a boundary layer of thickness similar to Ra-1/3 pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength similar to Ra-1/3 along the wall. As a result of the exact solution of the variational problem, the constant C is smaller than the previous estimate by a factor of 2.5 for no-slip and 1.6 for no-stress boundary conditions. This modest reduction in C indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55-70) are surprisingly accurate.