The Structure of Poloidal Fields Embedded in Thin Disks

TL;DR

The paper develops a global thin-disk model for embedding large-scale poloidal fields in magnetized accretion disks by closing the turbulent electromotive force with a tensorial diffusivity $D_{ijk}$ and a helicity-regulated dynamo tensor $\alpha_{ij}$. Magnetic helicity conservation links diffusion to dynamo action, producing a nonlinear backreaction that self-regulates the mean field and can even concentrate flux via anisotropic MRI turbulence. A physically motivated vertical boundary condition, $P_{\rm mag}=P_{\rm gas}$ with $\partial_z(B_r/B_\phi)=0$, connects the disk solution to an exterior force-free magnetosphere and removes degeneracy in stationary solutions. The results imply that thin disks can sustain bending angles of large magnitude, enabling magnetosphere-dominated angular momentum transport, and they provide a framework to interpret observed jet intermittency, magnetic rings, and polarized emission in a range of accreting systems. Overall, the work highlights the dynamic coupling between anisotropic turbulence, helicity regulation, and poloidal-field geometry in shaping disk–magnetosphere interactions.

Abstract

Many accreting systems are modeled as geometrically thin disks. Simulations of accretion disks cannot be extended to this regime, although local models can address the behavior of narrow annuli. A global model needs to account for the interactions between a large-scale poloidal field, accreted from the environment, and the disk. The disk magnetosphere can be modeled subject to the boundary conditions imposed by the disk. These depend on the structure of the magnetic field as it crosses the disk and the degree to which the disk can support a bend in the field lines. Building on earlier work we derive a set of equations describing a stationary disk with an embedded poloidal field. We derive a modified induction equation that incorporates tensorial turbulent diffusivities and a helicity-regulated $α$-effect. We quantify how helicity conservation introduces a nonlinear backreaction on the large-scale dynamo, dynamically coupling turbulent diffusion and $α$-quenching. We discuss the challenges encountered in finding a unique solution under stationary flows $E_φ=0$, which balances the inflow of $B_z$ due to accretion, the outflow due to radial diffusion of $B_z$, and the vertical movement of $B_r$ due to turbulent diffusion and buoyancy. The vertical profiles of both the azimuthal diffusion coefficient $D_{ijk}$ and the helicity-driven $α_{ij}$ demonstrate that changes in the radial gradient can restructure the magnetic field geometry. The ability of disks to sustain large bending angles in the poloidal field implies that angular momentum flux through the magnetosphere can dominate over internal transport even for weak fields. Competing factors can result in non-unique solutions, necessitating extra constraints and diagnostics that highlight the role of isotropic turbulence and helicity regulation in magnetized disk environments.

Figures (3)

• Figure 1: Left: Vertical profile of the azimuthal turbulent diffusion term $D_\phi(z)$ from Equation \ref{['eq:Diff_Azm']}, plotted for three values of the radial gradient of the vertical field: $\partial B_z/\partial{r} = 0.01$, $0.1$, and $0.3$. The diffusion term includes contributions from vertical derivatives of $B_r$ and $B_\phi$, and a radial derivative of $B_z$. As $\partial B_z/\partial{r}$ increases, the contribution from radial magnetic structure becomes more prominent, especially near the midplane. The magnetic field profile in the vertical direction is Gaussian, and in the azimuthal direction, it is sinusoidal. $B_r$ is derived from the gradient of $B_z$. All the turbulence parameters, Reynolds, Maxwell stress tensor, and anisotropy tensor, taken from direct and fitting simulation data Jacquemin-Ide+21. Right: Vertical profile of the helicity-regulated $\alpha_{rr}(z)$ coefficient for three values of the radial gradient of the vertical field, $\partial B_z/\partial{r} = 0.01$, $0.1$, and $0.3$. In this model, the poloidal component $B_r(z)$ is assumed to scale with both $\partial B_z/\partial{r}$ and $\partial_z B_z$, reflecting the coupling of radial and vertical magnetic structures in the induction equation. Larger $\partial B_z/\partial{r}$ values result in stronger poloidal fields, enhancing the helicity-driven backreaction and modulating the $\alpha$-effect accordingly.
• Figure 2: Vertical profiles of the azimuthal turbulent diffusion coefficient, $D_\phi(z)$ (Blue solid, left axis), and the helicity dynamo coefficient, $\alpha_{rr}(z)$ (purple dashed, right axis), for a vertically isothermal thin disk as describe in section \ref{['vertdisk']}. The x-axis is $z/H$. Curves show three imposed radial gradients of the mean vertical field, $\partial_r B_z=\{0.01,\,0.1,\,0.3\}$. $D_\phi$ implements equation(\ref{['eq:diffusion']}) with stresses $R_{ij}\propto \alpha P$ and $M_{ij}/\rho\propto \alpha c_s^{2}$, while $\alpha_{rr}$ follows from the helicity constraint, equation(\ref{['eq:Alpha_ij']}). We run the adapted parameters $\alpha=0.1$, $\tau_c=1$, $\bar{Q}_{rr}=0.7$, $\bar{Q}_{zz}=0.9$, $\bar{Q}_{\phi\phi}=0.1$, $\bar{Q}_{r\phi}=0.05$, and a toroidal field envelope $B_\phi/B_0=0.2$. Increasing $\partial_r B_z$ boosts $|D_\phi|$ near the midplane and drives larger $|\alpha_{rr}|$, illustrating the locking between transport and helicity predicted by the model as expected.
• Figure 3: Radial bending of the large-scale magnetic field at the thin disk surface. The $B_r/B_z$ as a function of the magnetization parameter $\beta_0^{-1} = B_z^2/(4\pi P_0)$ adapted the density profile as in Figure \ref{['fig:Dphi_alpha']} with different $H/R$ initial values.