Solution method and results

We solve eqn. 12 numerically by discretizing it on a logarithmic grid with 200 zones from $r_{\rm ms}=6$ to $r_{\rm out}=150$ with the dead-zone starting at $r_{\rm dead}=100$. Here and for the rest of this paper, radii will be given in units of gravitational radii $GM/c^2$. We treat the advective ( $\partial a/\partial t$) terms using the second order van Leer (1977) method. All other terms (including the principal part integral) are also differenced to second-order spatial accuracy. The time evolution is achieved through a simple first-order explicit scheme. To ensure numerical stability, we set the time-step to be $dt=(1/dt_{\rm ad}^2+1/dt_{\rm diff}^2+1/dt_{\rm field}^2)^{-1/2}$, where the advective, diffusive and field time-steps are given by $dt_{\rm ad}=0.5\min[\Delta r/(v+\eta_*/r)]$, $dt_{\rm diff}=0.5\min[\Delta r^2/\eta_*]$ and $dt_{\rm field}=0.5\min[h\Delta r/\pi \eta_*]$.

Figure 1 shows the time-evolution of $A_*$ for the case of ${P_{\rm m}}=2$ and various choices of $h/r$ from 0.01 to 0.16. In all cases, the flux threading the black hole grows from zero and achieves some positive steady state. In all cases, the final equilibrium flux threading the black hole exceeds $\pi r_{\rm ms}^2B_0$ (corresponding to $A_*=18B_0$), thereby establishing the basic fact that the plunge region can aid in the accumulation of significant magnetic flux through the black hole. For thicker disks, the increased inward advection of the field (due to the increased radial inflow speed of the accreting matter) coupled with the decreased effectiveness of field diffusion leads to significant enhancements of the black hole-threading flux above this baseline value. The dependence of the equilibrium value of $A_*$ on disk thickness and magnetic Prandtl number is shown in Fig. 1b. For small ${P_{\rm m}}$, the enhancement of the hole-threading flux above the canonical value of $A_*=18B_0$ is very small. However, for ${P_{\rm m}}$ of order unity or higher, there is a strong $h/r$-dependent enhancement.

Figure 1: Panel (a) : Time dependence of the black hole-threading flux for ${P_{\rm m}}=2$ and $h/r=0.01$ (magenta dot-dot-dot-dash line), $0.02$ (cyan dotted line), $0.04$ (blue dot-dash line), $0.08$ (red solid line), and $0.16$ (green dashed line). For comparison, $A_*/B_0=18$ corresponds to the flux of the uniform external field threading the radius of marginal stability. Time is in units of the viscous timescale at $r_{\rm ms}$, $t_{\rm visc}=r^2(R^3/GM)/\alpha h^2$. Panel (b) : Equilibrium value of $A_*/B_0$ as a function of $h/r$ for ${P_{\rm m}}=0.2$ (green dashed line), $2.0$ (red solid line) and $20.0$ (blue dotted line).

Figure 2: Magnetic field configuration for the initial condition (left panel), the final state of the ${P_{\rm m}}=2$, $h/r=0.08$ case (middle panel) and the final state of the ${P_{\rm m}}=20$, $h/r=0.08$ case (right panel). Note how the higher magnetic Prandtl number results in a powerful inward dragging of magnetic field and subsequent magnetization of the black hole. Each of these three panels is 50 gravitational radii ($50GM/c^2$) on a side.

The full magnetic field configuration can be derived by solving the potential problem for $a(r,z;t)$ using the solution method laid out in HPB. In Fig. 2, we show the initial field configuration as well as the final configuration for $h/r=0.08$ and two choices of magnetic Prandtl number ${P_{\rm m}}=2$ and $20$. The initial configuration deviates from a simple uniform field due to the fact that flux is excluded from the region $r<r_{\rm ms}$ which leads to a ``bowing'' of the field lines away from the radius of marginal stability. This curvature is rapidly reversed as field is advected inwards, finally achieving a steady state in which the bend angle of field lines as they enter the diffusive part of the disk is approximately constant. As pointed out by Lubow et al. (1994) and discussed below, we expect this bend angle (away from the disk normal) to be $i\sim\tan^{-1}(hP_{\rm m}/r)$. This is indeed seen in our equilibrium solutions.

The central quantity of interest in this work is the magnetic field threading the black hole event horizon. Recalling the definition of the flux function, it is straightforward to show that the magnetic field threading the event horizon is $B_{\rm H}=A_*/r_{\rm H}^2$ where $r_{\rm H}=2$ is the event horizon radius of the (slowly rotating) black hole considered in this work. From the results described above, we conclude that the equilibrium flux threading the black hole always exceeds the flux of the external uniform field through the plunge region ( $\pi r_{\rm ms}^2B_0$ corresponding to $A_*=18B_0$), sometimes by a large factor in the case of high effective magnetic Prandtl numbers and/or thick disks. Scaling to this fiducial flux, we have $B_{\rm H}=4.5\Upsilon B_0$, where $\Upsilon=A_*/18 B_0$. Using a least squares fit to the results displayed in Fig. 1, we find that a good approximation is $\Upsilon\approx 1+20{P_{\rm m}}(h/r)$. Hence, we have

$$B_{\rm H}\approx4.5\left[1+20{P_{\rm m}}\left(\frac{h}{r}\right)\right]B_0,$$ (17)

which is accurate to the 20% level for ${P_{\rm m}}<20$. As we discuss below, the factor multiplying the $P_{\rm m}h/r$ term in eqn. 17 has a dependence on the size of the dead zone; the precise form of eqn. 17 is strictly valid only for $r_{\rm dead}=100$.