[@2007NJPh....9..305B; @2014arXiv1412.0997B]:

$$\begin{aligned}
\mathcal{E}^{(\alpha)}{}_{\phi} & = & C_{S}\cos\theta\psi^{\left(-\right)}\left(r-r_{s}\right)\label{eq:babck}\\
 & \times & \int_{r_{b}}^{r}\frac{\psi^{\left(+\right)}\left(r'-r_{bt}\right)\!B\left(r',\theta\right)\!}{1+B^{2}\left(r',\theta\right)}\!d\!r'+{\displaystyle \frac{\overline{\chi}B\left(r,\theta\right)}{\overline{\rho}\ell^{2}}},\nonumber \end{aligned}$$

where $C_{S}$ is a free parameter that controls the strength of the $\alpha$- effect, the $\psi^{\left(\pm\right)}=\frac{1}{2}\left(1\pm\mathrm{erf}\left(100r\right)\right)$ are the Heaviside-like functions, with $r_{s}=0.95R$, $r_{bt}=0.725R$ and the $\overline{\chi}=\overline{\mathbf{a}\cdot\mathbf{b}}$ is helicity density of the fluctuating part of magnetic field with the $\mathbf{a}$ being a fluctuating vector-potential. [ Following results of @2014ApJ...791...59K we assume that the strength of the non-local $\alpha$-effect does not change significantly in the explored interval of the rotation rates.]{}

In our models the $\alpha$ effect takes into account the kinetic and magnetic helicities. [Conservation of the magnetic helicity results to the dynamical quenching of the $\alpha$-effect.]{} The helicity density of the fluctuating part of magnetic field, $\overline{\chi}$, is governed by the conservation law [@hub-br12]: $$\frac{\partial\overline{\chi}^{(tot)}}{\partial t}=-\frac{\overline{\chi}}{R_{m}\tau_{c}}-2\eta\overline{\mathbf{B}}\cdot\mathbf{\overline{J}}-\boldsymbol{\nabla\cdot}\boldsymbol{\boldsymbol{\mathcal{F}}}^{\chi},\label{eq:helcon-1}$$ where $\overline{\chi}^{(tot)}=\overline{\chi}+\overline{\mathbf{A}}\cdot\overline{\mathbf{B}}$ is the total magnetic helicity density and the $\boldsymbol{\boldsymbol{\mathcal{F}}}^{\chi}$ is the diffusive flux of the magnetic helicity, $R_{m}$ is the magnetic Reynolds number, and $\eta$ is the microscopic magnetic diffusivity.

The distribution of the turbulent parameters, e.g, [such as the typical convective turn-over time $\tau_{c}$, the mixing length $\ell$, the RMS convective velocity $u'$, the mean density $\bar\rho$ and its gradient $\mathbf{\boldsymbol{\Lambda}}^{(\rho)}=\boldsymbol{\nabla}\log\overline{\rho}$ are computed from the mixing-length model 