[Paper Review] Self-similar structure of magnetized ADAFs and CDAFs

Dong Zhang, Z. G. Dai|arXiv (Cornell University)|May 21, 2008

Astrophysics and Star Formation Studies4 citations

TL;DR

This paper extends self-similar models of magnetized advection-dominated accretion flows (ADAFs) and convection-dominated accretion flows (CDAFs) by incorporating a general three-component magnetic field (r, φ, z) in cylindrical coordinates. It shows that magnetic fields significantly alter disk structure: vertical fields suppress accretion and reduce sound speed, while toroidal fields increase inflow velocity and modify the α–αc relation, with strong fields enabling non-accreting solutions when α + gαc < 0.

ABSTRACT

(Abridged) We study the effects of a global magnetic field on viscously-rotating and vertically-integrated accretion disks around compact objects using a self-similar treatment. We extend Akizuki &amp; Fukue's work (2006) by discussing a general magnetic field with three components ($r, ϕ, z$) in advection-dominated accretion flows (ADAFs). We also investigate the effects of a global magnetic field on flows with convection. For these purposes, we first adopt a simple form of the kinematic viscosity $ν=αc_{s}^{2}/Ω_{K}$ to study magnetized ADAFs. Then we consider a more realistic model of the kinematic viscosity $ν=αc_{s}H$, which makes the infall velocity increase but the sound speed and toroidal velocity decrease. We next use two methods to study magnetized flows with convection, i.e., we take the convective coefficient $α_{c}$ as a free parameter to discuss the effects of convection for simplicity. We establish the $α_{c}-α$ relation for magnetized flows using the mixing-length theory and compare this relation with the non-magnetized case. If $α_{c}$ is set as a free parameter, then $|v_{r}|$ and $c_{s}$ increase for a large toroidal magnetic field, while $|v_{r}|$ decreases but $|v_ϕ|$ increases (or decreases) for a strong and dominated radial (or vertical) magnetic field with increasing $α_{c}$. In addition, the magnetic field makes the $α_{c}-α$ relation be distinct from that of non-magnetized flows, and allows the $ρ\propto r^{-1}$ or $ρ\propto r^{-2}$ structure for magnetized non-accreting convection-dominated accretion flows with $α+gα_{c}&lt; 0$ (where $g$ is the parameter to determine the condition of convective angular momentum transport).

Motivation & Objective

• To extend Akizuki & Fukue (2006) by including a general large-scale magnetic field with all three components (r, φ, z) in self-similar ADAF models.
• To investigate how global magnetic fields affect accretion flows with convection, particularly in CDAFs.
• To derive and analyze the α–αc relation in magnetized flows using mixing-length theory, contrasting it with non-magnetized cases.
• To examine the impact of different magnetic field geometries (radial, toroidal, vertical) on radial velocity, sound speed, and toroidal velocity in self-similar solutions.

Proposed method

• Uses self-similar treatment assuming flow variables depend only on radius r, neglecting vertical variation except in the z-momentum equation.
• Applies two viscosity models: ν = αcₛ²/ΩK and ν = αcₛH, to assess their differing effects on velocity and sound speed.
• Derives the Lorentz force in cylindrical coordinates using Alfven sound speeds, incorporating all three magnetic field components.
• Introduces the convective coefficient αc as a free parameter and derives the α–αc relation via mixing-length theory for magnetized flows.
• Simplifies equations under extreme limits (e.g., large βz or βφ) to obtain analytical solutions for key variables like c₁α.
• Compares results with non-magnetized CDAF models from NIA, focusing on structural and dynamical differences due to magnetic fields.

Experimental results

Research questions

• RQ1 How does a general three-component magnetic field (r, φ, z) alter the self-similar structure of magnetized ADAFs?
• RQ2 What are the effects of strong toroidal, radial, or vertical magnetic fields on radial inflow velocity, sound speed, and toroidal velocity in ADAFs?
• RQ3 How does the α–αc relation in magnetized CDAFs differ from that in non-magnetized CDAFs?
• RQ4 Under what conditions can magnetized CDAFs become non-accreting, and how does the magnetic field enable this?
• RQ5 How do different viscosity prescriptions (ν = αcₛ²/ΩK vs. ν = αcₛH) affect the resulting flow structure in magnetized disks?

Key findings

• A strong vertical magnetic field prevents disk accretion and reduces the isothermal sound speed, stabilizing the flow against collapse.
• Using ν = αcₛH increases the radial inflow velocity but decreases both the sound speed and toroidal velocity compared to the ν = αcₛ²/ΩK model.
• For large toroidal fields, increasing the convective coefficient αc leads to higher |vr| and |cs|, while strong radial fields reduce |vr| but increase |vφ|.
• The α–αc relation in magnetized flows differs significantly from the non-magnetized case, with magnetic fields enabling non-accreting solutions when α + gαc < 0.
• In the limit of strong vertical fields, the solution scales as c₁α ∼ 3(α + gαc)(1 + s)/[(1 − s)βz], showing strong suppression of accretion.
• For dominant radial fields, the solution is given by αc₁ = 2(α + gαc)[ε′′ + √(ε′′² + 2(α + gαc)²)]⁻¹, indicating a modified viscosity-convection coupling.