  Several evolved stars have been found to exhibit long-period radial velocity
variations that cannot be explained by planetary or brown dwarf companions.
Non-radial oscillations caused by oscillatory convective modes have been put
forth as an alternative explanation, but no modeling attempt has yet been
undertaken. We provide a model of a non-radial oscillation, aiming to explain
the observed variations of the cluster giant NGC 4349 No. 127. The star was
previously reported to host a brown dwarf companion, but whose existence was
later refuted in the literature. We reanalyzed 58 archival HARPS spectra,
acquiring additional activity indicators using the SERVAL and RACCOON
pipelines. We searched for periodicity in the indicators and correlations
between the indicators and radial velocities. We further present a simulation
code able to produce synthetic HARPS spectra, incorporating the effect of
non-radial oscillations, and compare the simulated results to the observed
variations. We find a positive correlation between chromatic index and radial
velocity, along with closed-loop Lissajous-like correlations between radial
velocity and each of the spectral line shape indicators (full width at half
maximum, and contrast of the cross-correlation function and differential line
width). Simulations of a low-amplitude, retrograde, dipole (l = 1, m = 1),
non-radial oscillation can reproduce the observed behavior and explain the
observables. Photometric variations below the detection threshold of the
available ASAS-3 photometry are predicted. The oscillation and stellar
parameters are largely in agreement with the prediction of oscillatory
convective modes. The periodic variations of the radial velocities and activity
indicators, along with the respective phase shifts, measured for the
intermediate-mass cluster giant NGC 4349 No. 127, can be explained by a
non-radial oscillation.
