  The temporal evolution of the fluid circulation generated by a buoyancy force
when two-dimensional (2D) arrays of 2D thermals are released into a quiescent
incompressible fluid is studied through the results of numerous lattice
Boltzmann simulations. It is observed that the circulation magnitude grows to a
maximum value in a finite time. When both the maximum circulation and the time
at which it occurs are non-dimensionalised by appropriately defined
characteristic scales, it is shown that two simple Prandtl number (Pr)
dependent scaling relations can be devised that fit these data very well over
nine decades of Pr spanning the viscous and diffusive regimes and six decades
of Rayleigh number (Ra) in the low Ra regime. Also, obtained analytically is
the exact result that circulation magnitude continues to grow in time for a
single buoyant vortex ring in an infinite unbounded fluid.
