A detailed description of the dynamo model can be found in our previous paper (Pipin & Kosovichev 2019, hereafter PK19). The model describes the dynamo generation of large-scale magnetic fields (LSMFs) in the bulk of the solar convective zone (CZ). The model is based on the mean-field induction equation (Krause & Rädler 1980):
1





where the induction vector of the LSMF, 





, is represented as the sum of the toroidal and poloidal components:





where r is the radial distance, θ is the polar angle, and 





 is the unit vector in the azimuthal direction. The mean electromotive force 





 describes the turbulent generation effects, pumping, and diffusion:
2





where the symmetric tensor αij stands for the turbulent generation of the LSMF by kinetic and magnetic helicities, the antisymmetric tensor γij describes the turbulent pumping effect, and the anisotropic (in the general case) tensor ηijk is the eddy diffusivity of the LSMF (Pipin 2018). The large-scale (LS) flow field, 





, produces the LS toroidal magnetic field from the LS poloidal field by means of the differential rotation, 





. The meridional circulation, 





, advects the LSMF in the convection zone. The angular momentum conservation and the equation for the azimuthal component of large-scale vorticity, 





, determine distributions of the differential rotation and meridional circulation:
3







4





where 





 is the turbulent stress tensor:
5





(see detailed description in PK19). Also, 





 is the mean density, 





 is the mean entropy, and 





 is the gradient along the axis of rotation. The mean heat transport equation determines the mean entropy variations from the reference state due to the generation and dissipation of LSMF and large-scale flows (Pipin & Kitchatinov 2000):
6





where 





 is the mean temperature, Fr is the radiative heat flux, and Fc is the anisotropic convective flux. An analytical mean-field expression for Fc takes into account the effect of the Coriolis force, and the influence of the LSMF on the turbulent convection (see PK19). The last two terms in Equation (6) take into account the convective energy gain and sink caused by the generation and dissipation of LSMF and large-scale flows. The reference profiles of mean thermodynamic parameters, such as entropy, density, and temperature are determined from the stellar interior model MESA (Paxton et al. 2011, 2013). The radial profile of the typical convective turnover time, τc, is determined from the MESA code, as well. We assume that τc does not depend on the magnetic field and global flows. The convective rms velocity is determined from the mixing-length approximation
7





where 





 is the mixing length, αMLT = 1.9 is the mixing-length parameter, and Hp is the pressure height scale. Equation (7) determines the reference profiles for the eddy heat conductivity, 





, eddy viscosity, νT, and eddy diffusivity, ηT, as follows:
8







9







10





The model gives the best agreement of the angular velocity profile with helioseismology results for 





 (PK19). Also, the dynamo model reproduces the solar magnetic cycle period, ∼22 years, if 





.
Citation: Pipin 2018
Citation function: Uses