  We study the Doppler-cooling of radial two-dimensional (2D) Coulomb crystals
of trapped barium ions in a radiofrequency trap. Ions in radial 2D crystals
experience micromotion of an amplitude that increases linearly with the
distance from the trap center, leading to a position-dependent frequency
modulation of laser light in each ion's rest frame. We use two tones of
Doppler-cooling laser light separated by approximately 100~MHz to efficiently
cool distinct regions in the crystals with differing amplitudes of micromotion.
This technique allows us to trap and cool more than 50 ions populating 4 shells
in a radial two-dimensional crystal, where with a single tone of Doppler
cooling light we are limited to 30 ions in 3 shells. We also individually
characterize the micromotion of all ions within the crystals, and use this
information to locate the center of the trap and to determine the Matthieu
parameters $q_{x}$ and $q_{y}$.
