Self-Similar Solutions for Viscous and Resistive ADAF

In this paper, the self-similar solution of resistive advection dominated accretion flows (ADAF) in the presence of a pure azimuthal magnetic field is investigated. The mechanism of energy dissipation is assumed to be the viscosity and the magnetic diffusivity due to turbulence in the accretion flow. It is assumed that the magnetic diffusivity and the kinematic viscosity are not constant and vary by position and $\alpha$-prescription is used for them. In order to solve the integrated equations that govern the behavior of the accretion flow, a self-similar method is used. The solutions show that the structure of accretion flow depends on the magnetic field and the magnetic diffusivity. As, the radial infall velocity and the temperature of the flow increase, and the rotational velocity decreases. Also, the rotational velocity for all selected values of magnetic diffusivity and magnetic field is sub-Keplerian. The solutions show that there is a certain amount of magnetic field that the rotational velocity of the flow becomes zero. This amount of the magnetic field depends on the gas properties of the disc, such as adiabatic index and viscosity, magnetic diffusivity, and advection parameters. The solutions show the mass accretion rate increases by adding the magnetic diffusivity and in high magnetic pressure case, the ratio of the mass accretion rate to the Bondi accretion rate decreases as magnetic field increases. Also, the study of Lundquist and magnetic Reynolds numbers based on resistivity indicates that the linear growth of magnetorotational instability (MRI) of the flow decreases by resistivity. This property is qualitatively consistent with resistive magnetohydrodynamics (MHD) simulations.

💡 Research Summary

In this work the authors develop a semi‑analytic model of an advection‑dominated accretion flow (ADAF) that incorporates both turbulent viscosity and turbulent magnetic diffusivity (resistivity) together with a pure toroidal magnetic field. Starting from the full set of magnetohydrodynamic (MHD) equations in spherical coordinates, they assume steady, axisymmetric flow confined to the equatorial plane and neglect self‑gravity and relativistic effects. The key novelty is the adoption of α‑prescriptions for both the kinematic viscosity ν and the magnetic diffusivity η that depend on the local gas pressure, density, Keplerian angular velocity Ω_K and on the magnetic pressure ratio β ≡ p_mag/p_gas. Specifically, ν = α p_gas/(ρ Ω_K)(1 + β)^{1‑μ} and η = η₀ p_gas/(ρ Ω_K)(1 + β)^{1‑μ}, where μ (0 ≤ μ ≤ 1) controls whether total pressure (μ = 0) or only gas pressure (μ = 1) enters the transport coefficients.

The authors seek self‑similar solutions of the form v_r ∝ r^{‑1/2}, Ω ∝ r^{‑3/2}, c_s² ∝ r^{‑1} and B_φ² ∝ r^{‑1}, introducing dimensionless constants c₁, c₂, c₃ that represent, respectively, the radial inflow speed, the angular velocity, and the ratio of sound‑to‑gravitational potential energy. A power‑law density profile ρ ∝ r^{s} is assumed, with s = ‑3/2 corresponding to a no‑wind configuration. Substituting these ansätze into the continuity, radial momentum, angular momentum, energy and induction equations yields a closed algebraic system (Eqs. 29‑31) linking c₁, c₂, c₃ to the physical parameters (α, η₀, β, γ, f, μ).

From this system the authors derive several important results:

1. Critical magnetic pressure β_b – The angular velocity vanishes when the composite parameter D₂ (which multiplies c₂²) becomes zero. Solving D₂ = 0 gives an analytic expression for a critical β, β_b = 18 α f η₀