Self-similar thermals in Stokes flow

Similarity solutions are obtained for the rise of a buoyant thermal in Stokes flow (i.e. infinite Prandtl number), in which all lengths scale like $t^{1/2}$ and velocities like $t^{-1/2}$. The dimensionless problem depends only on the Rayleigh number $Ra=B/(\nu\kappa)$, where $B$ is the (conserved) total buoyancy. For small $Ra$ there are only slight deformations to a spherically symmetric Gaussian temperature distribution. For large $Ra$ the temperature distribution is greatly elongated in the vertical direction, with a long `wake', which contains most of the buoyancy and dominates the flow, and a small `head', which is asymptotically unimportant. This structure contrasts with the usual view of mantle plumes and laboratory experiments. The large-$Ra$ behaviour is explained using a simple analytic model based on slender-body theory. The width of the thermal increases like $(\kappa t)^{1/2}$ while the wake length and rise height both increase like $(Ra\ln Ra)^{1/2}(\kappa t)^{1/2}$.