Magnetic fields in accretion discs
Marthijn de Kool, Geoffrey V. Bicknell, Zdenka Kuncic, PASA, 16 (3), 225.
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The vertical structure of magnetic accretion discs

The detailed vertical structure of accretion discs, as based on the equations of vertical hydrostatic equilibrium, energy transport, opacities and equation of state has been studied extensively in the past (e.g. Meyer & Meyer Hofmeister 1982, Mineshige & Osaki 1983, Canizzo & Wheeler 1984, Shaviv & Wehrse 1986). It is clear that when the magnetic effects discussed in the previous section are taken into account, the disc structure must be significantly different from those derived in these studies. A large part of the accretion luminosity may not be locally dissipated, as is assumed in the $\alpha$ heating prescription, but rather be transported out of the main body of the disc by buoyancy

Detailed numerical magnetohydrodynamical modelling of vertically stratified accretion discs with an isothermal or adiabatic equation of state has been performed by Stone et al. 1996. Their calculations neglected the energy transport and heating and cooling processes in the disc, and thus could not draw any conclusions regarding the formation of a hot corona, or compare their results to standard accretion disc models. They did find, in contrast to earlier analytical estimates and direct numerical simulations of the Parker instability (e.g Matsumoto & Shibata 1992), that buoyant transport in their models was very ineffective. One of the motivations for the present work was to investigate the reasons underlying the discrepancy between the result of Stone et al. and the other studies.

This work attempts to bridge the gap between the standard vertical structure models and the MHD calculations by including simplified terms describing the generation, dissipation and buoyant transport of magnetic field that (hopefully) catch the essence of the detailed MHD result in a detailed vertical structure calculation that can model the heating and cooling processes determining the structure of the accretion disc and the associated formation of a corona.

Equations and method

To solve for the detailed vertical disc structure we require a set of equations for the hydrostatic equilibrium, energy generation and transport, and magnetic field generation, dissipation and transport. We solve these equations treating the radiative transport in the grey two-stream approximation. The solution method is based on earlier work by Shaviv & Wehrse 1986 and Adam et al. 1988. The two-stream method approximates the full angle-dependent and frequency dependent radiation field by considering only an ingoing and an outgoing direction, and frequency averaged intensities. Although approximate, this method allows for a natural transition between optically thick and optically thin regions. This is not possible with the more standard way of solving the radiative transfer equation in diffusion approximation. Our treatment of the radiative energy transport has been described in more detail in de Kool & Wickramasinghe 1999.

The inclusion of magnetic fields in the vertical structure equations is a new ingredient, so we describe the equations used in more detail. We base ourselves on a physical interpretation of the results of Stone et al. 1996. The simple model described below should be seen as a parametric description, based on some physical arguments that hopefully make the results scale with two parameters in a reasonable way.

It is assumed that there is a local dynamo acting in the disc that creates magnetic energy density (or equivalently pressure) P_m at a rate $\Omega_K^{-1}$. Virtually every dynamo theory predicts a growth timescale of this order (e.g. Galeev, Rosner & Vaiana 1979), so this scaling is likely to be physically correct. However, when the magnetic field becomes too strong the Balbus-Hawley instability (Balbus & Hawley 1991), which performs an essential step in the dynamo mechanism by generating a radial magnetic field component from a vertical one, starts to be suppressed because the minimum wavelength of the instability $\lambda_{BH}$ becomes larger than the disc thickness. We model this suppression of magnetic field generation by multiplying the linear growth rate with a correction factor

$$A(x)= 0.74 \exp (-20(x-0.1)^3) , x = {{P_m}\over{P_g}}$$ (35)

which is numerically almost identical to the suppression factor originally derived by Tout & Pringle (1992)(their equation 2.2.3), except that it goes to zero exponentially rather than abruptly, something we require for numerical stability. Thus the generation of magnetic field is given by

$$\left({{d P_m}\over{dt}}\right)_{gen} = A(P_m/P_g) \Omega P_m$$ (36)

We assume that the dissipation (reconnection) rate scales with the typical length scale of the magnetic field variations divided by the Alfven speed. We take the typical length scale L to be $\ell h$, with h the disc scale height. The dissipation rate (and magnetic heating rate H_mag) is then given by

$$\left({{d P_m}\over{dt}}\right)_{dis} = - {{\gamma v_A} \over {\ell h}} P_m$$ (37)

where v_A is the Alfven speed, and $\gamma$ is an adjustable parameter, basically the reconnection speed divided by the Alfven speed. Note that in this description an equilibrium between generation and dissipation is reached because the generation is suppressed, not because the dissipation increases.

Finally, an equation for the buoyant transport of magnetic field is needed. We assume that the vertical flux of magnetic energy density is given by

$$F_B = v_{eq} \Delta P_m$$ (38)

where $\Delta P_m$ is the typical fluctuation in the magnetic energy density that gives rise to the upward/downward motion. v_eq is the speed at which a rising or falling element moves when the buoyant forces are in equilibrium with the frictional force exerted by the surroundings, defined by

$$\rho v_{eq}^2 = \Omega^2 z \Delta \rho L$$ (39)

with L the typical size of an element and $\Delta \rho$ the difference in the density in the element and the mean density enforced by pressure equilibrium,

$$\Delta \rho = - \rho {{\Delta P_m}\over{P_g}}$$ (40)

A simple consideration of the equation of motion of a rising element shows that it reaches this equilibrium speed by the time it has moved by less than its own size, so the approximation that the elements move at v_eq is reasonable.

Equations 38, 39 and 40 contain the two quantities $\Delta P_m$ and $\ell$, the values of which still have to be determined. To reduce the number of parameters of our model we argue that these two are related in the following way. In the turbulent disc we expect that the fluctuations in the pressure are of the order of the fluctuations in the turbulent momentum density,

$$\Delta (P_m+P_g) \sim \rho \Delta v ^2$$ (41)

The turbulence is driven by the differential rotation, so that we can make the estimate

$$\Delta v \sim L {{d v_K}\over{d r}}$$ (42)

with v_K the Keplerian velocity, and where it was implicitly assumed that typical sizes are the same in the radial and vertical direction. Combining these estimates, and using $h^2 \sim (c_s^2 + v_A^2)/{\Omega_K}^2$, this leads to

$$\Delta (P_m+P_g) \sim {1 \over 4} \ell^2 (P_m+P_g)$$ (43)

and we expect that roughly

$$\Delta (P_m) \sim {1 \over 4} \ell^2 (P_m)$$ (44)

Thus we are left with the parameter $\ell$, the ratio of the typical size of a region with enhanced or reduced magnetic field and the disc height, which determines the effectiveness of buoyant magnetic transport

$$F_B \sim {1 \over 8} \ell^{7 \over2} \Omega_K z^{1 \over 2} h^{1 \over 2} P_g^{-{1 \over 2}} P_m^{3 \over 2} \quad .$$ (45)

The numerical results of Stone et al. 1996 indicate that there is not much power in fluctuations on the larger scales comparable to the disc height, consistent with the result that they found buoyant transport to be very ineffective if we take the high power of $\ell$ in equation 45 into account. It is not clear, however, that the numerical diffusivity in their calculations properly models the reconnection processes that can combine smaller coherent regions of magnetic field into the larger ones for which magnetic buoyancy and escape can be important (Tout & Pringle 1992). We will therefore investigate a range of values for $\ell$ that covers both effective and ineffective buoyant transport.

Results

In this section we will compare the vertical structure of magnetic accretion discs with ineffective buoyant transport and with effective buoyant transport. The models are for an accretion disc around a 1 M_ M$_{\odot}$ compact object, at a radius of 3 x 10⁹ cm, and for 4 central temperatures: 10⁴, 3.5 x 10⁴, 6 x 10⁴ and 8 x 10⁴ K.

Figure 1: The vertical structure of a magnetic accretion disc with the parameters $\gamma =0.5$ and $\ell =0.5$, which is typical for the case in which buoyant magnetic energy transport is relatively inefficient. Figure 1a-b give the gas pressure and temperature as a function of height. Figure 1c gives the ratio of magnetic to gas pressure, and Figure 1d shows the radiative (solid) and buoyant magnetic (dashed) energy fluxes as a function of height. The cross indicates the height where the optical depth is unity.

Figure 2: The vertical structure of a magnetic accretion disc with the parameters $\gamma =0.1$ and $\ell =0.8$, for which buoyant magnetic energy transport plays a major role. Figure 2a-b give the gas pressure and temperature as a function of height. Figure 2c gives the ratio of magnetic to gas pressure, and Figure 2d shows the radiative (solid) and buoyant magnetic (dashed) energy fluxes as a function of height. The cross indicates the height where the optical depth is unity

Case I: Discs with predominantly local magnetic energy dissipation

As described in the previous section, our model contains the two parameters $\gamma$, the ratio of reconnection speed to Alfven speed and $\ell$, the ratio of the size of a magnetic region and the disc height. When $\gamma$ is large and $\ell$ small, reconnection is very efficient and most of the magnetic field is dissipated at the same place it is generated, before it has time to be transported by buoyancy effects. In figures 1a-d we present a set of models where this is the case, with the parameters $\gamma =0.5$ and $\ell =0.5$.

The inefficiency of buoyant transport in this case is best demonstrated in Figure 1d, which compares the vertical flux in radiation with that in buoyant magnetic field. For these parameters, the ratio of buoyant flux to radiative flux is about 0.1 deep inside the disc, ranges from 10^-1 to 10^-3 at $\tau_{R}= 1$ and falls to very small values at low optical depth.

In Figure 1c, the ratio of magnetic to thermal pressure is shown as a function of height. Deep inside the disc the dynamo mechanism regulates the magnetic pressure to be very close to the point where the wavelength of the BH instability is close to the disc height, with $P_m/P_g \sim 0.25$. As the pressure drops, the ratio of magnetic to thermal pressure increases to a maximum of 10-20. The generation of magnetic field is completely suppressed at this point, and the buoyant flux is being used up by dissipation, which becomes quite effective now because of the high Alfven speed. The magnetic field is dissipated so effectively that the ratio of magnetic to gas pressure actually starts to decrease again before the thermal instability point that represents our disc outer boundary is reached (de Kool & Wickramasinghe 1999). Two of the temperature profiles have a sharp maximum close to the outer edge, after which the radiative equilibrium temperature is regained once more before the thermal instability point is reached. This is caused by the sharp reduction in magnetic heating rate associated with the very sharp drop in P_m that is also evident from the decrease in P_m/P_g. This sharp reduction in heating rate allows the radiative equilibrium to be regained once more.

In Figure 1d we see the clear trend that the maximum in the buoyant flux occurs at higher optical depth as the central temperature (or equivalently the mass flux through the disc) is increased. For the models in figure 1, this leads to the result that for the lowest M ${\rm\dot M}$ a fraction of about 0.1 of the total flux is generated at low optical depth, presumably in the form of optically thin line emission, even though the disc as a whole is quite optically thick. For the highest T_c model presented this fraction is only about 10^-3.

Case II: Discs with efficient buoyant magnetic energy transport

In Figure 2a-d we present our results for the structure of discs in which buoyant magnetic energy transport plays a major role, as represented by a model with $\gamma =0.1$ and $\ell =0.8$. For these parameters, the buoyant flux deep inside the disc is about 4-6 times the radiative flux. At $\tau_{R}= 1$, the buoyant flux is still 4 times as large as the radiative flux for the lowest T_c model, and equal to the radiative flux for the highest T_c model, and in all cases the buoyant flux is still significant at the thermal instability point. The ratio of magnetic to gas pressure increases outwards as far as we can calculate, resulting in quite extended outer layers. In all cases, but especially the low T_c one, there is very significant dissipation in the optically thin but still relatively cool outer layers of the disc. The trend that the maximum in the buoyant flux occurs at higher optical depth as the mass flux through the disc is increased is even more obvious here than in Figure 1.

A fraction of 0.25 - 0.5 of the energy generated in the disc escapes past the thermal instability point, and will either escape to infinity in the form of Poynting flux, or will be dissipated beyond the instability point giving rise to a hot corona. However, our results show that this hot corona can not be in hydrostatic and thermal equilibrium and that dynamical effects such as outflows must become important. (See Meyer & Meyer-Hofmeister 1994 for a study of such outflowing coronae.)

Conclusions

The models indicate that buoyant magnetic transport can only be important if the magnetically over- and under-pressured regions have a size comparable to the disc scale height, and if the perturbation of the magnetic field is a significant fraction of the total pressure. Otherwise, their rise time is so long that reconnection even at a small fraction of the Alfven speed will dissipate the magnetic field before it can emerge. The hydromagnetic turbulence developing in the numerical MHD calculations of Stone et al. 1996 does not form such large coherent regions, and therefore these do not show significant buoyancy effects.

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