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
$Ra\rightarrow \infty$ 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
$Ra\rightarrow \infty$ limit and exhibiting solutions that achieve the ultimate
$Ra^{1/3}$ scaling. As expected, the optimizing flows have a boundary layer of thickness
${\sim}Ra^{-1/3}$ pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength
${\sim}Ra^{-1/3}$ 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.