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

There are several reasons why magnetic fields are thought to play an important role in accretion discs. It is the only source of viscosity that can be derived from first principles that comes close to being effective enough to explain the high observed viscosity of accretion discs. The existence of highly collimated jets in many disc accreting astronomical objects is most easily explained if they are driven by strong magnetic fields associated with the innermost disc. More recently, very hot accretion disc corona in which a significant part of the accretion luminosity is dissipated have been observed. This has led to many suggestions that the corona is heated by reconnecting magnetic field rising buoyantly from a cold accretion disc. A further development of this magnetically heated corona model is the main motivation for our work presented here. In this first section, we present a summary of recent work (Kuncic & Bicknell, 1999) and the reader is referred there for a complete analysis of the structure of turbulent magnetised discs.

The fundamental physical idea is that an accretion disc corona is formed in a similar fashion to the solar corona: by buoyant flux of magnetic field and subsequent magnetic heating. Some support for the picture of intense local dissipation when flux tubes reconnect comes from the work of Haardt & Maraschi 1997, who show that the coronal X-ray emission is best modelled if the corona is patchy. The magnetic field is generated in the accretion disc by dynamo action, at a rate of the same order as the gravitational power. If the magnetic fields in the accretion disk are responsible for the viscosity, they must be quite strong, roughly in equipartition with the gas pressure. Such a strong magnetic field will be buoyant, and is transported to several disc scale heights before it dissipates by reconnection in the high Alfven speed environment of the corona (Galeev et al. 1979, Tout & Pringle 1992). Note that magnetic heating of a corona is not the only possibility: the standard viscous heating assumption ( $H \propto \alpha P$) used in the Shakura & Sunyaev (1973) accretion disc models can also lead to hot outer disc layers since cooling in the low density environment is not very effective (Shaviv & Wehrse 1986, Meyer & Meyer-Hofmeister 1994). However, in these models most of the luminosity is still generated in the cold part of the disc where the pressure is high.

In the remainder of this section a careful development of the thin disc equations will be presented, with a turbulent magnetic field included. The formalism will include the possible effects of a disc wind and buoyancy, and will lead to an estimate of the buoyant Poynting flux in terms of the disc parameters.

Disc equations

In analogy with the Shakura-Sunyaev disc theory, we formulate vertically averaged thin disc equations by integrating the MHD equations between z = h and z = -h, where in our case h is the height of the disc-corona boundary. The surface density $\Sigma$ and average disc height h_av are defined by

$$\Sigma = \int_{-h}^{h} \rho dz \equiv 2 \overline{\rho} h$$ (1)

$$h_{av} = {2 \over \Sigma} \int_{0}^{h} \rho z dz \quad .$$ (2)

The mass flux in the disc wind and the mass flux through the disc are defined as

$$\dot M_w(r) = 4 \pi \int_{r_0}^{r} \rho^+ \ v_z^+ \ r\ dr$$ (3)

$$\dot M_a(r) = \dot M_a(r_0)\ + \ \dot M_w(r)$$ (4)

where r₀ is the innermost stable orbit, and the suffix ⁺ refers to conditions at the disc-corona boundary.

The radial momentum balance is described by the equation

$$\Sigma \left({{v_{\phi}^2}\over{r}} - {{G M}\over{r^2}} \rig... ...{\langle B_r B_z \rangle}\over{4 \pi}}\Bigr\vert _{-h}^{h} = 0$$ (5)

which leads to the expression for the azimuthal velocity

$$v_{\phi}^2 = v_{K}^2 - {{2 r \langle B_r^+ B_z^+ \rangle}\over{4 \pi \Sigma}}$$ (6)

where $v_{\rm K} = (GM/R)^{1/2}$ is the local Keplerian velocity. The second term on the RHS of this equation represents the radial force due to the magnetic tension along field lines penetrating the disc surface. If the magnetic field strength is not significantly above equipartition, this term can be shown to be $\mathrel{<\kern-1.0em\lower0.9ex\hbox{$\sim$}}(h/r) v_K^2$ so that the rotation is still close to Keplerian.

For the vertical momentum balance we ignore the dynamical terms, and obtain

$${{\partial P_g}\over{\partial z}} + {{\partial}\over{\partia... ...\langle B_r B_z \rangle \right) + {{G M}\over{r^3}} \rho z = 0$$ (7)

Integrating this equation from z=0 to z=h yields

$$P_{\rm g}(r,0) + {{\langle B_r^2 + B_{\phi}^2 - B_z^2 \rangle}\over{8\pi}}(r,0) = {1 \over 2}{{G M}\over{r^3}}\Sigma h_{av}$$ (8)

where $h_{av} = \int_0^h \rho z \, dz/\int_0^h \rho \, dz$ is an average scale height based upon the profile of the mean density. Because of the strong shearing we expect $B_{\phi}$ to dominate the magnetic energy density and to good approximation $P_{tot} = P_g + {{B_{\phi}^2}\over{8 \pi}}$. For an approximately isothermal disk

$$(P,\rho) = (P_0, \rho_0) \, exp(-z^2/h_s^2)$$ (9)

where P₀ and $\rho_0$ are the central pressure and density, respectively, and $h_{\rm s}$ is the scale-height. We use this to approximately relate $h_{\rm av}$ and h_s, thus, $h_{\rm s} = \sqrt \pi h_{\rm av}$. The following theory does not depend strongly upon this relationship; it simply improves upon the obvious order of magnitude relationship $h_{\rm s} \sim h_{\rm av}$.

Using this relationship between h_s and $h_{\rm av}$, together with $\Sigma(r) \approx \sqrt \pi \rho_0 h_{\rm s}$, and taking $v_{\rm s}$ as the isothermal sound speed, we can solve equation (8) for the scale-height:

$$\frac {h_{\rm s}}{r} = \sqrt 2 \frac{v_s}{v_K} \, \left[ 1 + \frac{B^2}{8 \pi P_0} \right]^{1/2}.$$ (10)

The angular momentum transport through the disc is described by the vertically averaged $\phi$ component of the momentum equation

$${d \over {dr}}[\dot M_a r v_{\phi} + r^2 h(r) \langle \overl... ...\langle {B_{\phi} B_{z}} \rangle}\over 2}\Bigr\vert _{-h}^{h}$$ (11)

The first term on the RHS represents angular momentum loss from the disc surface due to a wind, the second term angular momentum loss due to tension along magnetic field lines crossing the surface. If there is no angular momentum loss from the surface the RHS of this equation is zero, and we derive the following expression for the total averaged turbulent stress $\overline{\tau}_{r \phi}$, which is composed of magnetic and hydrodynamic turbulent contributions:

$$\bar \tau_{r \phi} = {{\langle \overline{B_r B_{\phi}}\rangl... ...rangle = {{\dot M_a v_{\phi}}\over{4 \pi r h}} [1 - \zeta(r)]$$ (12)

Here $\zeta(r)$ is set by the boundary condition specifying the stress at r₀. For the usual assumption of vanishing stress at r=r₀, we have $\zeta(r)=\sqrt{r_0/r}$.

Shearing box studies of MHD turbulence in accretion discs (Stone et al. 1996) have shown that the hydrodynamic and magnetic stresses are comparable. Therefore, it is convenient to parametrize these stresses in the following way:

$$\langle \overline{\rho v_r' v_{\phi}'} \rangle = -(1 - \eta) \bar \tau_{r \phi}$$ (13)

$${{\langle \overline{B_r B_{\phi}}\rangle}\over{4 \pi}} = \eta \bar \tau_{r \phi}$$ (14)

with $\eta \sim 0.5$. To make the connection with non-magnetic standard $\alpha$-disc models, we parametrise the total stress in term of the gas pressure, $P_{\rm g}$:

$$\langle \overline{\rho v_r^\prime v_{\phi}^\prime} \rangle = \alpha \overline{P}_{\rm g}$$ (15)

As in standard Shakura-Sunyaev discs, we expect $\alpha \mathrel{<\kern-1.0em\lower0.9ex\hbox{$\sim$}}1$.

We now turn to the energy equations. The overall disc energy balance is expressed by

$$L_D = \int_{r_0}^{r} 4 \pi r \sigma T_{eff}^4 dr = {{G M \do... ...le B_{\phi}^+ B_z^+}\over{4 \pi}}\right] dr + \xi L_{c} + L_Q$$ (16)

The terms on the RHS have the following physical meaning. The first one is the standard disc luminosity. The second represents gravitational potential energy lost in a wind. The next two magnetic terms are the Poynting flux lost in the wind and the work done against the disc by magnetic tension. The last two terms are the fraction $\xi$ of the coronal luminosity L_c that is absorbed by the disc and the conductive energy flux from corona into the disc L_Q.

The full energy equation contains many processes that we do not fully understand. It is therefore convenient to parametrize them in terms of the total available gravitational power,

$$P_G = {{G M \dot M_a(r_0)}\over{2 r_0}}$$ (17)

We write the luminosity of the disc as

$$L_D = (1 - f_w - f_B) P_G + \xi L_{c} + L_Q$$ (18)

If all the energy lost from the disc through the magnetic terms is used to heat the corona, we have

L_c = (1 - q_c) f_B P_G (19)

L_Q = q_c f_B P_G (20)

where q_c is the fraction of the coronal power that is conducted back into the disc.

High energy ($\sim$ 100 keV) spectra of Seyfert galaxies can generally be fitted successfully with a thermally Comptonised spectrum with a typical temperature and optical depth of the scattering gas of $T \sim 10^9$ K and $\tau_{es} \sim 1$, or a Compton y-parameter of order 1. Using the parametrization above, we find

$$e^y - 1 = {{L_c}\over{L_D}}= {{(1 - q_c) f_B}\over{1 - f_w - f_B[1 - \xi (1-q_c) - q_c]}}$$ (21)

Most disc corona models do not take all possible effects into account, e.g. Haardt & Maraschi 1997 assume f_w = q_c =0

Magnetic field evolution

To describe the evolution of the turbulent magnetic field in the accretion disc, we start with the general induction equation

$${{\partial{\bf B}}\over{\partial t}} + {\rm curl}({\bf B \times v}) = {{c^2}\over{4 \pi \sigma}} {\bf\nabla^2 B}$$ (22)

>From this we can derive the equation describing the generation of magnetic energy density $\epsilon_B$,

$${{\partial \epsilon_B}\over{\partial t}} + {{\partial}\over{... ... = {{B_i B_j}\over{4 \pi}} s_{ij} + {\rm dissipation \ terms}$$ (23)

Here s_ij is the shear tensor, $s_{ij}= {1\over 2}\bigl[ v_{i,j} + v_{j,i} - {2 \over 3} v_{k,k} \delta_{ij}\bigr]$. This identifies a mean volume rate of generation of magnetic energy density, given by

$$\dot \epsilon_B = {{\langle B_i B_j \rangle}\over{4 \pi}} \l... ...\frac {3 \eta G M \dot M_{\rm a}}{8 \pi r^3} \, (1 - \zeta(r))$$ (24)

where we have used the expression for the stress in terms of the mass accretion rate (equation (12)). Thus, for significant values of $\eta$, the rate of magnetic energy generation is comparable to the gravitational power. Galeev, Rosner & Vaiana 1979 have argued that this energy is mostly dissipated in a corona because dissipation inside the disc is negligible as a result of the low Alfven speed (see also the next section).

Similarly, the rate of generation of turbulent kinetic energy is given by:

$$\dot \epsilon_{TKE} = - \langle \rho v_i' v_j' \rangle \lang... ... {3 (1- \eta) G M \dot M_{\rm a}}{8 \pi r^3} \, (1 - \zeta(r))$$ (25)

and this reduces to the standard, unmagnetised disk expression, when $\eta=0$. We emphasize that neither of these expressions directly give the rate of dissipation of energy as is conventionally assumed in accretion disk theory. This involves another step, either dissipation at the dissipative scale of a turbulent cascade of a turbulent cascade or dissipation resulting from reconnection of transported energy or both.

Poynting flux resulting from buoyancy

In order to estimate the power that may be dissipated in the corona, we extend the above theory to consider the buoyant transport of magnetic field in a disc. We balance the drag force on a flux tube of diameter D, located a height z above the central plane of the disk and with stronger than average magnetic field, with the buoyancy force in the following way:

$$C_{\rm D} \, \rho v_{\rm b}^2 \, D \approx \left( \frac {\de... ...(- \frac {\partial P}{\partial z} \right) \, \frac{\pi D^2}{4}$$ (26)

where $C_{\rm D}$ is the drag coefficient and $v_{\rm b}$ is the buoyant velocity. This gives:

$$\Rightarrow v_{\rm b} \approx \left( \frac {\pi}{4} \right)^... ...2} \right)^{1/2} \, \left( \frac {h_s}{r} \right) \, v_{\rm K}$$ (27)

(Compare equation (39) in section 2). Equation (27) for the buoyant velocity leads to an estimate of the buoyant rise time, $t_{\rm b}$ compared with the Keplerian time $t_K = \Omega_K^{-1}$ of

$$\frac{t_{\rm b}}{t_{\rm K}} = \left( \frac {\pi}{4 C_D} \rig... ...rho}{\rho} \right)^{-1/2} \, \left( \frac {z}{D} \right)^{1/2}$$ (28)

Consideration of this eqaution at a typical scale-height, $z \sim h_{\rm s}$, we see that in order that the flux tube not be disrupted by shearing processes which occur on a Keplerian timescale, before it buoyantly rises out of the disk, we require $\delta \rho /\rho \sim 1$ and $D \sim h_s$. (This comparison may be somewhat over-restrictive, since shearing instabilities for an azimuthal field grow less rapidly than for a perpendicular field.)

If the flux tube is not disrupted, its rise through the accretion disk leads to a Poynting flux of electromagnetic energy. Using our estimate of the buoyant velocity we can then determine the ratio of Poynting flux into the corona to the total generation rate of energy per unit area. First, the Poynting flux, S_z perpendicular to the disk, is given by:

$$\approx \frac {B^2}{4 \pi} \, v_{\rm b}$$ S_z

$$\approx \left( \frac {\pi}{2} \right)^{1/2} \, C_{\rm D}^{-1/2} \, \left(... ...h_{\rm s}^2}\right) \, \left( \frac {B_0^2}{4 \pi P_0} \right) \, P_0 v_{\rm s}$$ (29)

where we have used equation (27) for the buoyant velocity and equation (10) for the scale-height and scaled the magnetic energy density by the mid-plane pressure. Two fundamental disk parameters enter this expression in the form of the product $P_0 v_{\rm s}$. The central pressure, P₀ can be estimated from the average disk pressure, $\bar P$, and the $\alpha$ relation, viz.,

$$\frac {2}{\sqrt \pi} \, \frac {h}{h_{\rm s}} \, \bar P$$ P₀ = (30)

$$\bar P \alpha^{-1} (1-\eta) \, \frac {\dot M_{\rm a} v_{\rm K}}{4 \pi r h} \, \left[1 - \zeta(r) \right]$$ =

$$\alpha^{-1} (1-\eta) \, \frac {\dot M_{\rm a} v_{\rm K}}{4 \pi r^... ...ight)^{-1} \, \left(\frac {h_{\rm s}}{r}\right)^{-1} \left[1 - \zeta(r) \right]$$ = (31)

(see equations (12) and (15)).

The expression for $\bar P$ introduces another factor of $h_{\rm s}/r$ which is absorbed in the expression for the isothermal sound speed derived from equation (10) for the scale-height, viz.,

$$v_{\rm s} = \frac {1}{\sqrt 2} \left( \frac {h_{\rm s}}{r} \... ..., v_{\rm K} \, \left( 1 + \frac{B_0^2}{8 \pi P_0} \right)^{-1}$$ (32)

Combining equations (29), (30) and (32) gives for the Poynting flux

$$S_z \approx \alpha^{-1} (1-\eta) C_D^{-1/2} \, \frac {GM \do... ...2} \right) \, \frac {B^2/8\pi P_0}{(1 + B_0^2/8\pi P_0)^{1/2}}$$ (33)

and the ratio of Poynting flux to the rate of generation of turbulent and magnetic energy per unit area of one side of the disk ( $h (\dot \epsilon_B + \dot \epsilon_{\rm TKE})$; see equations (24 and (e:e_turb)) is given by:

$$f_B \approx \frac{2}{3} \alpha^{-1} \frac {1-\eta}{\eta} \, ... ...2} \right) \, \frac {B^2/8\pi P_0}{(1 + B_0^2/8\pi P_0)^{1/2}}$$ (34)

Note that the magnetic field enters in two different ways in this expression, the first through the local value (B) and the second through the value at the disc mid-plane (B₀). It is apparent from this equation, that for say, $\alpha \sim 0.1$, $\delta \rho /\rho \sim 1$, $C_D \sim 1$, $D \sim h_{\rm s}$ and $B^2 / 8 \pi P_0 \sim 0.1$ at $z = h_{\rm s}$, the Poynting flux estimated here is an appreciable fraction of the rate of turbulent and magnetic energy generation. This gives some support to the notion that the generation of magnetic field in the disc, followed by its transport into the corona where it is dissipated, is an attractive means of producing a hot corona. Another factor which diminishes the Poynting flux is the filling factor of such flux tubes within the disc. Clearly this has to be of order unity, for buoyant transport to be important.

It must be acknowledged that the details of buoyant transport of magnetic energy are not well understood and are indeed controversial. The next section represents an attempt at understanding some of the details of magnetic transport in accretion discs.

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