  It has been shown recently that magnetic twist and axisymmetric MHD modes are
ubiquitous in the solar atmosphere and therefore, the study of resonant
absorption for these modes have become a pressing issue as it can have
important consequences for heating magnetic flux tubes in the solar atmosphere
and the observed damping. In this investigation, for the first time, we
calculate the damping rate for axisymmetric MHD waves in weakly twisted
magnetic flux tubes. Our aim is to investigate the impact of resonant damping
of these modes for solar atmospheric conditions. This analytical study is based
on an idealized configuration of a straight magnetic flux tube with a weak
magnetic twist inside as well as outside the tube. By implementing the
conservation laws derived by \cite{Sakurai:1991aa} and the analytic solutions
for weakly twisted flux tubes obtained recently by \cite{Giagkiozis:2015apj},
we derive a dispersion relation for resonantly damped axisymmetric modes in the
spectrum of the Alfv\'{e}n continuum. We also obtain an insightful analytical
expression for the damping rate in the long wavelength limit. Furthermore, it
shown that both the longitudinal magnetic field and the density, which are
allowed to vary continuously in the inhomogeneous layer, have a significant
impact on the damping time. Given the conditions in the solar atmosphere,
resonantly damped axisymmetric modes are highly likely to be ubiquitous and
play an important role in energy dissipation.
  We also suggest that given the character of these waves, it is likely that
they have already been observed in the guise of Alfv\'{e}n waves.
