  We present a harmonic expansion of the observed line-of-sight velocity field
as a method to recover and investigate spiral structures in the nuclear regions
of galaxies. We apply it to the emission-line velocity field within the
circumnuclear starforming ring of NGC1097, obtained with the GMOS-IFU
spectrograph. The radial variation of the third harmonic terms are well
described by a logarithmic spiral, from which we interpret that the
gravitational potential is weakly perturbed by a two-arm spiral density wave
with inferred pitch angle of of 52+/-4 degrees. This interpretation predicts a
two-arm spiral distortion in the surface brightness, as hinted by the dust
structures in central images of NGC1097, and predicts a combined one-arm and
three-arm spiral structure in the velocity field, as revealed in the
non-circular motions of the ionised gas within the circumnuclear region of this
galaxy. Next, we use a simple spiral perturbation model to constrain the
fraction of the measured non-circular motions that is due to radial inflow. We
combine the resulting inflow velocity with the gas density in the spiral arms,
inferred from emission line ratios, to estimate the mass inflow rate as a
function of radius, which reaches about 0.011 Msun/yr at a distance of 70 pc
from the center. This value corresponds to a fraction of about 4.2 x 10^{-3} of
the Eddington mass accretion rate onto the central black hole in this
LINER/Seyfert1 galaxy. We conclude that the line-of-sight velocity not only can
provide a cleaner view of nuclear spirals than the associated dust, but that
the presented method also allows the quantitative study of these possibly
important links in fueling the centers of galaxies, including providing a
handle on the mass inflow rate as a function of radius.
