the near-surface toroidal and radial magnetic fields (including the phase relation between them) as discussed in our previous papers (see, e.g, @2014ApJ_pipk).]{} For the B-type model we use parameters to make results for the solar case close to that discussed by @2011AstL...37..286K.

The numerical integration is carried out in latitude from pole to pole, and in radius from $r_{b}=0.715R_{\odot}$ to $r_{e}=0.99R_{\odot}$. The numerical scheme employs the pseudo-spectral approach for the numerical integration in latitude and the finite second-order differences in radius. The initial field was a weak dipole type poloidal magnetic field. The numerical scheme preserves the parity of the initial field unless there is a real parity braking process, e.g., associated with non-symmetric, relative to the equator, perturbations of the dynamo parameters. The initial conditions represent a weak dipole field with strength 0.01G.

For each rotation rate from the set $\left\{ 0.85,1,1.25,1.5,1.6,1.75\right\} \times\Omega_{0}$, where $\Omega_{0}=2.87\times10^{-6}$s$^{-1}$ we study the different nonlinear dynamo regimes. The models D0 and B0 is related to the dynamo with the “algebraic” quenching of the $\alpha$-effect. The model D0 includes the buoyancy effect as well. In increasing the number of the model we include the other nonlinear effects, such as, the magnetic helicity and the magnetic 