  We study the nucleation and dynamics of vortices in rotating lattice
potentials where weakly linked condensates are formed with each condensate
exhibiting an almost axial symmetry. Due to such a symmetry, the on-site phases
acquire a linear dependence on the coordinates as a result of the rotation,
which allows us to predict the position of vortices along the low density paths
that separate the sites. We first show that, for a system of atoms loaded in a
four-site square lattice potential, subject to a constant rotation frequency,
the analytical expression that we obtain for the positions of vortices of the
stationary arrays accurately reproduces the full three-dimensional
Gross-Pitaevskii results. We then study the time-dependent vortex nucleation
process when a linear ramp of the rotation frequency is applied to a lattice
with sixteen sites. We develop a formula for the number of nucleated vortices
which turns to have a linear dependence on the rotation frequency with a
smaller slope than that of the standard estimate which is valid in absence of
the lattice. From time-dependent Gross-Pitaevskii simulations we further find
that the on-site populations remain almost constant during the time evolution
instead of spreading outwards, as expected from the action of the centrifugal
force. Therefore, the time-dependent phase difference between neighboring sites
acquires a running behavior typical of a self-trapping regime. We finally show
that, in accordance with our predictions, this fast phase-difference evolution
provokes a rapid vortex motion inside the lattice. Our analytical expressions
may be useful for describing other vortex processes in systems with the same
on-site axial symmetry.
