Dependence on the size of the dead zone

At first glance, the dragging of magnetic flux by the accretion disk leads to a surprisingly large enhancement in the black hole-threading field. However, as we will now explain, simple arguments can be put forward to support the results encapsulated in eqn. 17.

Firstly, we note that the existence of the dead zone is crucial for setting an overall size scale to the magnetic disturbances introduced by the disk. To see this, consider the limit in which $r_{\rm dead}\rightarrow \infty$ (also requiring $r_{\rm out}\rightarrow \infty$, of course). In this case, the imposed uniform magnetic field is dragged inwards by the accretion flow but a balance will never be achieved between the inward advection and the magnetic tension -- without an imposed spatial scale, the field curvature through the disk and hence the magnetic tension can be made arbitrarily small. A balance is possible only when one imposes an outer truncation on the part of the disk that drags the magnetic flux. In this case, the undragged field at $r>r_{\rm dead}$ acts as an anchor and limits the vertical extent to which the magnetic field can be appreciable distorted. Indeed, our calculations show that the magnetic field at $\vert z\vert>r_{\rm dead}$ is essentially just the imposed uniform field.

Now, as already noted, we find that the magnetic field threads the active part of the diffusive accretion disk ( $r_{\rm ms}<r<r_{\rm dead}$) with a bend angle (away from the disk normal) of $\tan i \equiv B_r/B_z\approx hP_{\rm m}/r$. As shown by HPB and Lubow et al. (1994), this is a direct consequence of a balance between outward magnetic diffusion due to field-line tension and the inwards advection of magnetic field,

$$v_r\frac{\partial A}{\partial r}\approx\eta_*\left(\frac{\partial A}{\partial z}\right)_{z=h}.$$ (18)

Consider the field line that threads the inner edge of the diffusive disk at $r=r_{\rm ms}$. This field line follows a roughly parabolic path in the magnetosphere that can be described by the flux function $\Psi=\Psi_0(r^2+2\xi z)={\rm constant}$. We can determine the parameter $\xi$ using the fact that, at the disk plane, we have $B_r/B_z\approx r/hP_{\rm m}$,

$$\frac{B_r}{B_z}=-\frac{\partial \Psi/\partial z}{\partial \P... ...rac{\xi}{r_{\rm ms}}\approx \left(\frac{h}{r}\right)P_{\rm m},$$ (19)

where we have dropped a term that is second order in $(h/r)$. At a vertical distance of $z=r_{\rm dead}$, this same field line has a cylindrical radius $R$ given by

$$R^2=r_{\rm ms}^2\left[1+2\frac{r_{\rm dead}}{r_{\rm ms}}\left(\frac{h}{r}\right)P_{\rm m}\right]$$ (20)

Using our observation above concerning the vertical extent of the field disturbances, we use the fact that the field is essentially uniform for $\vert z\vert>r_{\rm dead}$ to read off the magnetic flux threading the plunge region and hence the black hole,

$$\Phi_{\rm H}=\pi R^2 B_0=\pi r_{\rm ms}^2 B_0\left[1 +2\frac... ...rm dead}}{r_{\rm ms}}\left(\frac{h}{r}\right)P_{\rm m}\right].$$ (21)

In terms of the field threading the hole (putting $r_{\rm ms}=6$) we get

$$B_{\rm H}=4.5\left[1 +\frac{r_{\rm dead}}{3}\left(\frac{h}{r}\right)P_{\rm m}\right]B_0.$$ (22)

Thus we can see that the numerical factor multiplying the $(h/r)P_{\rm m}$ term in eqn. 17 is directly related to the value of $r_{\rm dead}$.

Figure 3: Magnetic field configurations for the plunge boundary condition (left panel) and the uniform flux bundle boundary condition. In both cases, the figure shows a zoom-in ($10GM/c^2$ on a side) of the final field structure in the ${P_{\rm m}}=2$, $h/r=0.08$ case. A white vertical line on the accretion disk denotes the radius of marginal stability.

The above discussion helps to elucidate the role of the plunge region in enhancing the black hole-threading flux -- the plunge region ``shields'' the diffusive part of the disk from the large bundle of magnetic flux that threads the black hole. This bundle of flux is the ultimate repository for the magnetic flux that has been scooped up the by accretion flow. The larger the region of the disk that can drag the flux inwards, the larger is this repository. To illustrate this issue, we have run a modified version of our code in which the plunge region boundary condition is replaced with the assumption that the magnetic flux contained within $r=r_{\rm ms}$ has the form of a uniform field on the disk plane. We employ canonical values of the model parameters; $h/r=0.08$, $P_{\rm m}=2$, and $r_{\rm dead}=100$. As expected, we get a weak (50%) enhancement in the flux contained within $r=r_{\rm ms}$, compared with over a factor of 3 for the plunge case. The magnetic field structures of the two cases are illustrated in Fig. 3.

Performing a full numerical solution to eqn. 12 for $r_{\rm dead}=50$, $r_{\rm dead}=100$ and $r_{\rm dead}=200$ reveals that the enhancement of the magnetic flux increases with $r_{\rm dead}$ slightly more slowly than the linear relationship predicted by our simple arguments in this section. Since the implementation of the dead zone is one of most artificial aspects of our toy model, we will not explore this dependence in any more detail in this paper. In real systems, the dead zone might be identified with the outer edge of the MHD turbulence dominated accretion disk, e.g., the self-gravity region in an AGN disk or the tidal truncation radius for the disk in a Galactic Black Hole Binary (GBHB). Both of these radii are likely to be at significantly larger radius than $r_{\rm dead}=100$ used here. Alternatively, if the magnetosphere is treated using a full MHD wind model, the crucial length-scale which determines the magnetic field enhancement is likely be the vertical height of the Alfvénic surface. It is beyond the scope of this paper to address such models. However, our approach allows us to illustrate an essential point; that the inward dragging of magnetic field over some region of the inner disk coupled with the existence of the plunge region allows a significant enhancement in the strength of the magnetic field threading the black hole.