Standing torsional Alfv\'en waves as the source of the rotational period variation in magnetic early-type stars

Context. The influence of magnetic fields on stellar evolution remains unresolved. It has been proposed that if there is a large-scale magnetic field in the stellar interior, torsional waves could arise, efficiently transporting angular momentum. In fact, the observed variations in the rotation periods of some magnetic stars may be attributed to these torsional waves' standing waves. Aims. To demonstrate the existence of torsional waves through modeling of the rotational period variations. Method. We conduct an eigenmode analysis of standing waves based on one-dimensional magnetohydrodynamic equations. The internal magnetic field structures are parametrically represented to treat poloidal fields with different degrees of central/surface concentration. The obtained frequencies are compared with the observed frequencies of the rotational period variations, thereby constraining the internal magnetic field structures. Results. The 67.6 years exhibited by CU Vir is reproduced for surface-concentrated magnetic field structures. The rotational period variations of all ten magnetic stars analyzed in this study are inconsistent with a centrally concentrated magnetic field. Conclusions. Torsional waves can reproduce the observations of rotational period variations. The large-scale magnetic fields within magnetic stars would be concentrated on the surface.