  We analyze propagation of acoustic vortex beams in longitudinal synthetic
magnetic fields. We show how to generate two field configurations using a fluid
contained in circulating cylinders: a uniform synthetic magnetic field hosting
Laguerre-Gauss modes, and an Aharonov-Bohm flux tube hosting Bessel beams. For
non-paraxial beams we find qualitative differences from the well-studied case
of electron vortex beams in magnetic fields, arising due to the vectorial
nature of the acoustic wave's velocity field. In particular, the pressure and
velocity components of the acoustic wave can be individually sensitive to the
relative sign of the beam orbital angular momentum and the magnetic field. Our
findings illustrate how analogies between optical, electron, and acoustic
vortex beams can break down in the presence of external vector potentials.
