Comparison between axisymmetric numerical magnetohydrodynamical simulations and self-similar solutions of jet-emitting disks

Abstract

Turbulent accretion disks threaded by a large-scale vertical field near equipartition can drive tenuous and fast self-confined jets. Self-similar solutions of these jet-emitting disks (JEDs) have been known for a long time and provide the distributions of all physical quantities, from the turbulent disk to the asymptotic regime of ideal magnetohydrodynamic (MHD) jets. However, a thorough comparison with time-dependent numerical simulations has never been achieved, mostly because mass-loss rates found in simulations were always larger than those found analytically. This tension may have cast doubt on the analytical approach, the numerical one, or both. Our goal is to bridge the gap between these two complementary approaches and settle this long-standing issue. We performed 2.5D (axisymmetric) simulations of resistive and viscous accretion disks described by the same parameter sets as analytical JED solutions. The results demonstrate an almost perfect agreement between the numerical and analytical solutions, thereby resolving the previously observed tension. The simulations also confirm that JEDs behave as dynamical attractors: starting from different initial conditions, the system consistently converges toward the expected steady-state solution. This work demonstrates that self-similar solutions provide valuable insights into accretion-ejection physics. However, as 2.5D numerical simulations which rely on alpha-prescriptions, they strongly depend on the assumptions made for turbulent terms. In contrast, 3D simulations capture the turbulence, but become prohibitively expensive when modeling large-scale astrophysical systems. We advocate for the use of global 3D simulations to investigate turbulence and to derive physically motivated prescriptions for use in 2.5D studies.

Figures (16)

• Figure 1: Two-dimensional visualizations at $t = 1000$ for simulations with $\alpha_P = 0$, $1$, and $2$, from left to right. The top panels display the full computational domain, while the bottom panels show a zoomed-in view of the inner region. Dashed lines, with increasing opacity, represent the slow magnetosonic (SM), Alfvén, and fast magnetosonic (FM) critical surfaces, respectively. White lines indicate magnetic-field lines anchored at $R = 2, 10, 20, 30$, and up to $90$. The background color map shows the density in arbitrary units.
• Figure 2: Time evolution of the anchoring radius $R_{ss}$ of the last magnetic field line intersecting the FM surface for each simulation. The dashed curves represent the final stationary radii: $R_{ss} \!=\! 13.3$ for $\alpha_P \!=\! 0$, $R_{ss} \!=\! 4.5$ for $\alpha_P \!=\! 1$ and $R_{ss} \!=\! 2.3$ for $\alpha_P \!=\! 2$.
• Figure 3: Radial distributions of disk accretion rate, $\dot{M}_{acc}$; magnetic flux, $\Psi$; and magnetization, $\mu$; measured at the disk midplane ($\theta \!=\! \pi/2$) for each simulation at $t \!=\! 1000$. Dashed lines indicate simple power-law fits as functions of the cylindrical radius, $r$ (see text for details).
• Figure 4: Latitudinal profiles of thermal pressure $P$; laminar (total) magnetic pressure, $P_{lam}$; turbulent magnetic pressure, $P_{turb}$ (all normalized to $P$ at the disk midplane); components of the magnetic field $B_R$, $B_{\theta}$, and $B_{\phi}$ (normalized to $B_z$ at the disk midplane), and components of the velocity $v_R$, $v_{\theta}$, and $v_{\phi}$ (normalized to the Keplerian velocity $v_{K}$ at the disk midplane), measured at $R \!=\! 2.0$ and at $t \!=\! 1000$ for each simulation. The gray shaded area on the right corresponds to the disk, the white area to the disk wind, and the gray shaded area on the left to the spine.
• Figure 5: Profiles of MHD invariants along magnetic-field line anchored at $R \!=\! 2.0$ for each simulation at $t \!=\! 1000$. The quantities shown are the mass-loading parameter, $\kappa$; the normalized angular velocity of the magnetic surface, $\omega_*$; the magnetic lever arm, $\lambda$; and the normalized total specific energy, $e$. The shaded area indicates the resistive MHD region, while the dashed vertical lines mark the positions of the SM, Alfvénic, and FM critical surfaces.