  Red supergiants are cool massive stars and are the largest and the most
luminous stars in the universe. They are characterized by irregular or
semi-regular photometric variations, the physics of which is not clearly
understood. The paper aims at deriving the velocity field in the red supergiant
star $\mu$ Cep and relating it to the photometric variability with the help of
the tomographic method. The tomographic method allows to recover the
line-of-sight velocity distribution over the stellar disk and within different
optical-depth slices. The method is applied to a series of high-resolution
spectra of $\mu$ Cep, and these results are compared to those obtained from 3D
radiative-hydrodynamics CO5BOLD simulations of red supergiants. Fluctuations in
the velocity field are compared with photometric and spectroscopic variations,
the latter being derived from the TiO band strength and serving (at least
partly) a proxy of the variations in effective temperature. The tomographic
method reveals a phase shift between the velocity and spectroscopic/photometric
variations. This phase shift results in a hysteresis loop in the temperature -
velocity plane, with a timescale of a few hundred days, similar to the
photometric one. The similarity between the hysteresis loop timescale measured
in $\mu$ Cep and the timescale of acoustic waves disturbing the convective
pattern suggests that such waves play an important role in triggering the
hysteresis loops.
