  A geometric phase is found to arise from the cyclic adiabatic variation of
the crossed magnetic and electric fields which sustain the Brillouin rotation
of a plasma column. The expression of the gauge field associated with this
geometric phase accumulation is explicited. The physical origin of this phase
is shown to be the uncompensated inductive electric field drift that stems from
magnetic field cyclic variations. Building on this result, the effect of a
weak, periodic and adiabatic modulation of the axial magnetic field on the
particle guiding center drift motion is demonstrated to be equivalent to that
of a perpendicular electric field, allowing to study the gauge induced
Brillouin flow through a geometrically equivalent linear radial electric field.
This finding opens new perspectives to drive plasma rotation and hints at
possible applications of this basic effect.
