Magnetostatic equilibrium of the accretion disks of the T Tauri stars

The problem of the magnetostatic equilibrium of an accretion disk with a large-scale magnetic field is solved. The equation of the magnetostatic equilibrium is written taking into account the gravity, the gas and magnetic pressure gradients. The heat transfer equation takes into account heating by the dissipation of turbulence and cooling by the radiation. The ordinary diffrential equations of the model are solved by the Runge — Kutta method of the $4$th order of accuracy. The radial structure of the disk is simulated using the accretion disk model of Dudorov and Khaibrakhmanov.
An analytical solution of the induction equation for the azimuthal component of the magnetic field, $B_\varphi$, shows that the $B_\varphi(z)$ profile, generally speaking, is nonmonotonic when the magnetic field is assumed to be symmetric about the equatorial plane and the Dirichlet's boundary condition is used on the surface of the disk. If the Neumann's boundary condition is used at the surface, then $B_\varphi$ increases monotonically with the height.
Numerical calculations show that, in the first case, the vertical gradient of the magnetic pressure leads to a thickening of the disk if the magnetic Reynolds number $R_{\rm m}\gg 1$, the density and temperature profiles become flatter, and the photosphere is located higher than in the case of no magnetic field. In the second case, the magnetic field causes the disk to «compress». The deviation of the disk height from the hydrostatic one is of $10$–$15~\%$. A dynamically strong magnetic field is generated outside the «dead» zone (a region of a low degree of the ionization), which extends from $r=0.3$ au up to $(10$–$20)$ au with typical parameters for a T Tauri star of the solar mass. Possible observed manifestations of the revealed features of the vertical structure of a disk with a magnetic field are discussed.