  We analyze a three-dimensional rapidly rotating convection model of tall
columnar structure in the limit of infinite Prandtl number, i.e., when the
momentum diffusivity is much more dominant than the thermal diffusivity.
Consequently, the dynamics of the velocity field takes place at a much faster
time scale than the temperature fluctuation, and at the limit the velocity
field formally adjusts instantaneously to the thermal fluctuation. We prove the
global well-posedness of weak solutions and strong solutions to this model.
