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Red supergiants are cool massive stars and are the largest and the mostluminous stars in the universe. They are characterized by irregular orsemi-regular photometric variations, the physics of which is not clearlyunderstood. The paper aims at deriving the velocity field in the red supergiantstar $\mu$ Cep and relating it to the photometric variability with the help ofthe tomographic method. The tomographic method allows to recover theline-of-sight velocity distribution over the stellar disk and within differentoptical-depth slices. The method is applied to a series of high-resolutionspectra of $\mu$ Cep, and these results are compared to those obtained from 3Dradiative-hydrodynamics CO5BOLD simulations of red supergiants. Fluctuations inthe velocity field are compared with photometric and spectroscopic variations,the latter being derived from the TiO band strength and serving (at leastpartly) a proxy of the variations in effective temperature. The tomographicmethod reveals a phase shift between the velocity and spectroscopic/photometricvariations. This phase shift results in a hysteresis loop in the temperature -velocity plane, with a timescale of a few hundred days, similar to thephotometric one. The similarity between the hysteresis loop timescale measuredin $\mu$ Cep and the timescale of acoustic waves disturbing the convectivepattern suggests that such waves play an important role in triggering thehysteresis loops.

Tomography of cool giant and supergiant star atmospheres II. Signature of convection in the atmosphere of the red supergiant star $\mu$ Cep