2 Formation of the super-Keplerian layer

Angular momentum in standard disks (Shakura & Sunyaev 1973, hereforth SS-disks) is transported outwards mainly via small scale magneto-hydrodynamical turbulence. Magnetic fields (-MFs) however were assumed to be weak, and remain so during the whole viscous evolution of the disk. Here we adopt a different view. Let $\nu_{\rm tot}= \nu_{\rm HD} + \nu_{\rm Mag} = \alpha (P_{\rm gas} + \pi_{\rm B})/\Omega$be a modified dynamical viscosity (see the caption of Fig. 1 for elaboration). Unlike $\nu_{\rm HD}$ which transports angular momentum outwards, $\nu_{\rm Mag}$ transports angular momentum in the vertical direction, provided that $B_{\rm P}$ is of large scale topology. Taking into account that $\nu_{\rm HD}$ in SS-disks decreases inwards, a transition radius  $R_{\rm Tr}$ at which result. the time scale of angular momentum removal by MFs at $R_{\rm Tr}$ is of the same order as that by the turbulent viscosity. Hence

$$\frac{\tau_{\rm rem}}{\tau_{\rm tur}} \simeq \alpha \left(\fr... ...{r}\right)\left(\frac{V_{\rm s}}{V_{\rm A}}\right)^2 \simeq 1,$$ (1)

where ${V_{\rm s}},~V_{\rm A}$ are the sound and transverse $\rm Alf\grave{v}en$ speeds $(V_{\rm A}=\sqrt{B_{\rm P} B_{\rm T}/4 \pi \rho})$, respectively. Taking into account that ${H}/{r} \ge \alpha$, we end up with ${V_{\rm A}} \ge \alpha{V_{\rm s}}.$The last optimistic inequality applies everywhere in the disk, irrespective whether turbulence is mediated by  $\pi_{\rm B}$or not. Therefore, vertical transport of angular momentum via MFs is at least as efficient as $\alpha-$viscosity. This is even more justified by the fact that Balbus-Hawley instability in Keplerian-disks amplifies initially weak fields to considerably large values, but remain still below equipartition (Hawley et al. 1996). Imposing appropriate boundary conditions, and carrying the MHD-Box calculations with high spatial resolution, MFs could be amplified up to equipartition (Ziegler 2002), yielding thereby ${\tau_{\rm rem}} < {\tau_{\rm tur}}$. Furthermore, the dynamo-action model proposed by Tout & Pringle (1992), if applied, would make ${\tau_{\rm rem}}$ even shorter.

To be noted here that when taking a more realistic density and temperature stratification in global 3D MHD disk-calculations, vertical transport of angular momentum is inevitable (Arlt 2002).

Efficient vertical transport of angular momentum rises the following important issues: 1) Turbulence in the disk need not be dissipated, or it might be even suppressed by the amplified $B_{\rm P}$. This allows accretion to evolve without necessarily emitting the bulk of the their potential energy as radiation, and gives rise to energy re-distribution. 2) Accretion flows may not proceed as slowly as in SS-disk, but they may turn into advection-dominated. This occurs because the time scale of angular momentum removal from the disk scales as:

$$\tau_{\rm rem} \sim \rho V_{\rm T} H/B_{\rm P} B_{\rm T} \sim r^{3/2},$$ (2)

where $V_{\rm T} (=r \cos{\theta}~ \Omega)$ is the angular velocity (Fig. 1). This indicates that angular momentum removal is more efficient at smaller radii.

To maintain dynamical stability, angular momentum removal from the disk should be compensated by rapid advection from larger radii, i.e., $\tau_{\rm adv} =\tau_{\rm rem}$. Further, rapid and steady generation of $B_{\rm T}$ in the disk yields $B_{\rm P}/B_{\rm T} = H_{\rm d}/r.$ The later two conditions imply that the radial velocity $U_{\rm r} \sim V_{\rm A}$, which means that the stronger the MF threading the disk, the more advection-dominated it becomes, and therefore the faster is the establishment of the super-Keplerian layer.

Based on the present calculations (see the caption of Figs. 1 and  2-5), it is found that: 1) Angular velocity in the transition layer (TL) adopts approximately the profile $\Omega \sim r^{-5/4}$. 2) Energy dissipation is injected primarily into the ions that cool predominantly through fast outflows. 3) The generated toroidal magnetic field is quenched by a magnetic diffusion (reconnection) and fast outflows. The width of the TL is pre-dominantly determined through the transverse variation of the ion-pressure across the jet, i.e., $H_{\rm W} = P_{\rm i}/\nabla P_{\rm i} \approx 0.2~r$, and so strongly dependent on whether the flow is a one- or two-temperature plasma. In the steady-state case, this implies:

$$\rho \sim r^{-7/4}, T_{\rm i} \sim r^{-1/2}, U_{\rm r} \sim r^{-1/4}, B_{\rm P}/B_{\rm T} \sim \rm const.$$ (3)

We note that $U_{\rm r}$ adopts a profile and attains values similar to those in the innermost part of the disk. Provided that energy exchange between the matter in the disk and in the TL is efficient, the incoming matter can easily be re-directed into outwards-oriented motions. This implies that the Bernoulli number (Be) can change sign in dissipative flows. As Fig. 5 shows, Be is everywhere negative save the TL, where it attains large positive values, so that the ion-plasma can start its kpc-journey. Worth-noting is the resulting MF topology (see Fig. 4). Apparently, the outflow is sufficiently strong to shift the MF lines outwards, while the large diffusivity prevents the formation of large electric currents along the equator. In the corona however, MFs are too weak to halt the diffusive plasma in the dynamically unstable corona against gravity, and instead, they drift with the infalling gas inwards. In the case of very weak MFs ( $\beta \le 0.1$), our calculations indicate a considerably weak outflow. This is a consequence of the tendency of the MFs to establish a monopole like-topology, i.e, a one-dimensional MF-topology in which ${B}_\theta \longrightarrow0$. In this case, the magnetic tension $\pi_{\rm B}$ becomes inefficient in feeding the matter in the TL with the angular momentum required for launching jets, indicating herewith that cold accretion disks alone are in-appropriate for initiating winds (Ogilvie & Livio 2001).

Figure 2: The horizontal distribution of the normalized density $\rho$, angular velocity $\Omega$, radial and horizontal velocities $U_{\rm r},~V_{\rm z},$ the radial and horizontal MF-components $(B1, B2)=(B_{\rm r}, B_\theta )=B_{\rm P}$ and the toroidal MF-component $B_{\rm T}$ at r=2.5. Note the density plateau, the positive radial velocity (outflows), the super-Keplerian rotation and the strongly enhanced strength of the MF-components in the TL. The inwards-oriented motions in the disk (inflows) and strongly increasing density towards the equator are obvious.

Comparing the flux of matter in the wind to that in the disk, we find that ${{~\dot{\cal M}}}_{\rm W}/{{~\dot{\cal M}}}_{\rm d} = {\rm const.} \approx 1/20$. The angular momentum flux associated with the wind is $\dot{\cal J}_{\rm W}/\dot{\cal J}_{\rm d} = {a}~ ({{~\dot{\cal M}}}_{\rm W}/{{~\dot{\cal M}}}_{\rm d})~r^{1/4},$ where a is a constant of order unity. Consequently, at $r = 3\times 10^2\ R_{\rm Sch}$, almost $25\%$ of the total accreted angular momentum in the disk re-appears in the wind.

Why is the TL geometrically thin?
In stratified dissipative flows the density scale height is much smaller than the scale height of the angular velocity (Hujeirat & Camenzind 2000). Since the flow in TL rotates super-Keplerian, Coriolis forces act to compress the disk-matter and make its density scale height even smaller. This implies that advective-disks are geometrically thin, much thinner than what ADAF-solutions predict.
On the other hand, unless there is a significant energy flux that heats up the plasma from below, as in the case of stars, heat conduction will always force the BH-coronae to collapse dynamically. To elaborate this point, let us compare the conduction time scale with the dynamical time scale along $B_{\rm P}$-field at the last stable orbit of a SMBH:

$$\frac{\tau_{\rm cond}}{\tau_{\rm dyn}} = \frac{r \rho U_{\rm ... ... 4.78\times 10^{-4} \rho_{10} T^{-5/2}_{\rm i,10} {\cal M}_8,$$ (4)

where $\rho_{10}$, ${T_{\rm i,10}}$ and ${\cal M}_8$ are respectively in $10^{-10}~\rm {g~cm^{-3}}$, 10¹⁰ K and in $10^8~{~{\cal M}_{\odot}}$ units. This is much less than unity for most reasonable values of density and temperature typical for AGN-environments. In writing Eq. (4) we have taken optimistically the upper limit $c/\sqrt{3}$ for the velocity, and set  $\kappa_0 = 3.2\times 10^{-8}$ for the ion-conduction coefficient. When modifying the conduction operator to respect causality, we obtain ${\tau_{\rm cond}}/{\tau_{\rm dyn}} \le U_{\rm r}/c$, which is again smaller than unity.
This agrees with our numerical calculations which rule out the possibility of outflows along the rotation axis, and in particular not from the highly unstable polar region of the BH, as ADAF-solutions predict.

Figure 3: The distribution of the velocity field superposed on the logarithmic-scaled ratio of the ion- to electron-temperatures (red color corresponds to high ratios and blue to low-ratios). The lower figure is a zoom-in of the flow configuration in the innermost part of the disk.