Critical Rayleigh number of for error function temperature profile with a quasi-static assumption

When a semi-infinite body is heated from below by a sudden increase in temperature (or cooled from above) an error function temperature profile grows as the heat diffuses into the fluid. The stability of such a profile is investigated using a large-wavelength asymptotic expansion under the quasi-static, or frozen-time, approximation. The critical Rayleigh number for this layer is found to be $Ra=\pi^{1/2}$ based on the length-scale $(\kappa t)^{1/2}$ where $\kappa$ is the thermal diffusivity and $t$ the time since the onset of heating.