Formation of Thermal Vortex Rings

An evolution of a spherical region, subjected to uniform buoyancy force, is investigated. Incompressibility and axial symmetry are assumed, together with a buoyancy discontinuity at the boundary. The boundary turns into a vortex sheet and the system evolves into a ring. Contrary to the case of mechanically generated rings, buoyancy-driven rings are unstable. This is due to the generation of negative vorticity at the bottom. Furthermore, a sequence of Kelvin-Helmholtz instabilities arises along the buoyancy anomaly boundary. This sequence transfers the energy toward large scales with $\kappa^{-3}$ distribution. The vortex blob method has been used to simulate the system numerically. An optimization algorithm, used previously in two dimensions, has been extended to the axisymmetric case. It reduces computational complexity from $N^2$ to $N \log N$, where N is the number of nodes. Additionally, a new algorithm has been developed as a remedy for the exponential growth of the number of nodes required. It exploits a tendency of the vortex sheet to form many parallel stripes, by merging them together.