Weak magnetic field, solid-envelope rotation, and wave-induced N-enrichment in the SPB star $\zeta$ Cassiopeiae

Aims. The main-sequence B-type star $\zeta$ Cassiopeiae is known as a N-rich star with a magnetic field discovered with the Musicos spectropolarimeter. We model the magnetic field of the star by means of 82 new spectropolarimetric observations of higher precision to investigate the field strength, topology, and effect. Methods. We gathered data with the Narval spectropolarimeter installed at T\'elescope Bernard Lyot (TBL, Pic du Midi, France) and applied the least-squares deconvolution technique to measure the circular polarisation of the light emitted from $\zeta$ Cas. We used a dipole oblique rotator model to determine the field configuration by fitting the longitudinal field measurements and by synthesizing the measured Stokes V profiles. We also made use of the Zeeman-Doppler Imaging technique to map the stellar surface and to deduce the difference in rotation rate between the pole and equator. Results. $\zeta$ Cas exhibits a polar field strength $B_{\rm pol}$ of 100-150 G, which is the weakest polar field observed so far in a massive main-sequence star. Surface differential rotation is ruled out by our observations and the field of $\zeta$ Cas is strong enough to enforce rigid internal rotation in the radiative zone according to theory. Thus, the star rotates as a solid body in the envelope. Conclusions. We therefore exclude rotationally-induced mixing as the cause of the surface N-enrichment. We discuss that the transport of chemicals from the core to the surface by internal gravity waves is the most plausible explanation for the nitrogen overabundance at the surface of $\zeta$ Cas.