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Aims. The main-sequence B-type star $\zeta$ Cassiopeiae is known as a N-richstar with a magnetic field discovered with the Musicos spectropolarimeter. Wemodel the magnetic field of the star by means of 82 new spectropolarimetricobservations of higher precision to investigate the field strength, topology,and effect.  Methods. We gathered data with the Narval spectropolarimeter installed atT\'elescope Bernard Lyot (TBL, Pic du Midi, France) and applied theleast-squares deconvolution technique to measure the circular polarisation ofthe light emitted from $\zeta$ Cas. We used a dipole oblique rotator model todetermine the field configuration by fitting the longitudinal fieldmeasurements and by synthesizing the measured Stokes V profiles. We also madeuse of the Zeeman-Doppler Imaging technique to map the stellar surface and todeduce the difference in rotation rate between the pole and equator.  Results. $\zeta$ Cas exhibits a polar field strength $B_{\rm pol}$ of 100-150G, which is the weakest polar field observed so far in a massive main-sequencestar. Surface differential rotation is ruled out by our observations and thefield of $\zeta$ Cas is strong enough to enforce rigid internal rotation in theradiative zone according to theory. Thus, the star rotates as a solid body inthe envelope.  Conclusions. We therefore exclude rotationally-induced mixing as the cause ofthe surface N-enrichment. We discuss that the transport of chemicals from thecore to the surface by internal gravity waves is the most plausible explanationfor the nitrogen overabundance at the surface of $\zeta$ Cas.

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