Description: Investigating the influence of surface friction on the inertial oscillation (IO) of an extratropical, non-growing, convective boundary layer (CBL), we paid particular attention to the stability-dependent interactive coupling of shear-induced turbulence and turbulent friction, which leads to a nonlinear relationship between friction and velocity. We showed that in contrast to common perception, IO damping is controlled not only by friction but also by the dependence of friction on velocity. Furthermore, we found that surface friction not only causes damping but also modifies the restoring force. Using these basic principles, we studied the oscillatory properties (equilibrium, periodicity and damping) of the CBL by means of a model based on Monin–Obukhov surface-layer similarity (MOS) and the mixed-layer approximation. We found that the model complies with a quadratic surface stress–velocity relationship (QS) in the neutral limit, and a linear surface stress–velocity relationship (LS) in the proximity of the free-convective limit. Dynamically, the LS leads to a harmonic oscillation with a constant periodicity and exponential damping of the IO. However, the QS displays rather complex anharmonic behaviour; in comparison with the LS it produces a 50% stronger overall damping and a 100% larger contribution to the restoring force. Considering CBLs of arbitrary stability, we found that the MOS stress–velocity relation can be very well approximated by a much simpler linear combination of the LS and the QS which, respectively, represent the convective and the shear-induced contributions to friction. This enabled us to link the set of the external parameters (surface roughness, surface buoyancy flux and boundary layer depth) to a set of three effective parameters: the equilibrium velocity, the convective friction constant and the neutral friction constant. Together with the Coriolis coefficient, these parameters completely determine the IO. Copyright © 2012 Royal Meteorological Society