The propagation of a high energy laser through a nearly stagnant
absorbing medium is studied. The absorption values and time scale of
the problem are such that the laser induces convective heat currents
transverse to the beam. These currents couple to the laser via the
refractive index, causing time dependent thermal blooming. A numerical
method is developed and applied to the model in [J. Electromagn. Waves
Appl. 33, 96
(2019)JEWAE50920-507110.1080/09205071.2018.1528183], using
radial basis functions for spatial differencing, which allows for
irregular point spacings and a wide class of geometries. Both the beam
and laser-induced fluid dynamics are numerically simulated. These
simulations are compared to a historical experiment of a 300 W
laser in a smoke-filled chamber with good agreement; both cases
include a crescent shaped spot at the target.