In earlier works we pointed out that the disk's surface layers are non-turbulent and thus highly conducting (or non-diffusive) because the hydrodynamic and/or magnetorotational (MRI) instabilities are suppressed high in the disk where the magnetic and radiation pressures are larger than the plasma thermal pressure. Here, we calculate the vertical profiles of the {\it stationary} accretion flows (with radial and azimuthal components), and the profiles of the large-scale, magnetic field taking into account the turbulent viscosity and diffusivity and the fact that the turbulence vanishes at the surface of the disk. Also, here we require that the radial accretion speed be zero at the disk's surface and we assume that the ratio of the turbulent viscosity to the turbulent magnetic diffusivity is of order unity. Thus at the disk's surface there are three boundary conditions. As a result, for a fixed dimensionless viscosity $\alpha$-value, we find that there is a definite relation between the ratio ${\cal R}$ of the accretion power going into magnetic disk winds to the viscous power dissipation and the midplane plasma-$\beta$, which is the ratio of the plasma to magnetic pressure in the disk. For a specific disk model with ${\cal R}$ of order unity we find that the critical value required for a stationary solution is $\beta_c \approx 2.4r/(\alpha h)$, where $h$ the disk's half thickness. For weaker magnetic fields, $\beta > \beta_c$, we argue that the poloidal field will advect outward while for $\beta< \beta_c$ it will advect inward. Alternatively, if the disk wind is negligible (${\cal R} \ll 1$), there are stationary solutions with $\beta \gg \beta_c$.
Analysis of the diffusion and advection of a large-scale magnetic field in a accretion disk with a turbulent viscosity and magnetic diffusivity arising from the magnetorotational instability (MRI) shows that a weak large-scale field diffuses outward rapidly (van Ballegooijen 1989;Lubow, Papaloizou, & Pringle 1994). We mention but do not consider here the opposite limit where the magnetic field is sufficiently strong that it suppresses the MRI instability so that the disk is non-turbulent but accretion occurs due to angular momentum outflow to a magnetic disk wind or jet (Lovelace, Romanova, & Newman 1994). Earlier, Bisnovatyi-Kogan and Lovelace (2007) pointed out that the disk's surface layers are highly conducting because the MRI instability is suppressed in this region where the magnetic and radiative energy-densities are larger than the thermal gas energy-density. Rothstein and Lovelace (2008) analyzed this problem in further detail and discussed the connections with global and shearing box magnetohydrodynamic (MHD) simulations of the MRI. Lovelace, Rothstein, & Bisnovatyi-Kogan (2009;hereafter LRBK) developed an analytic model for the vertical (z) profiles of the stationary accretion flows (with radial and azimuthal components), and the profiles of the large-scale, magnetic field taking into account the turbulent viscosity and diffusivity due to the MRI and the fact that the turbulence vanishes at the surface of the disk.
Here, we require that the radial accretion speed be zero at the disk’s surface, and we assume that the ratio of the turbulent viscosity to the turbulent magnetic diffusivity is of order unity as suggested by MHD shearing-box simula-tions (Guan & Gammie 2009). For a fixed dimensionless viscosity α-value, we find that there is a definite relation between the ratio R of the accretion power going into magnetic disk winds to the viscous power dissipation and the midplane plasma-β, which is the ratio of the plasma to magnetic pressure in the disk.
Section 2 discusses the model for the flow and ordered magnetic field in a viscous diffusive disk. Section 3 discusses the solutions for a specific disk model. Section 4 gives the conclusions.
Following LRBK we consider the non-ideal magnetohydrodynamics of a thin axisymmetric, viscous, resistive disk threaded by a large-scale dipole-symmetry magnetic field B. We use a cylindrical (r, φ, z) inertial coordinate system in which the time-averaged magnetic field is B = B r r+B φ φ +B z ẑ, and the time-averaged flow velocity is
(2) These equations are supplemented by the continuity equation, ∇ • (ρv) = 0, by ∇ × B = 4πJ/c, and by ∇ • B = 0. Here, η is the magnetic diffusivity, F ν = -∇•T ν is the viscous force with T ν jk = -ρν(∂v j /∂x k +∂v k /∂x j -(2/3)δ jk ∇• v) (in Cartesian coordinates), and ν is the kinematic viscosity. For simplicity, in place of an energy equation we consider the adiabatic dependence p ∝ ρ γ , with γ the adiabatic index.
We assume that both the viscosity and the diffusivity are due to magneto-rotational (MRI) turbulence in the disk so that
where P is the magnetic Prandtl number of the turbulence assumed a constant of order unity (Bisnovatyi-Kogan & Ruzmaikin 1976), α ≤ 1 is the dimensionless Shakura-Sunyaev (1973) parameter, c s0 is the midplane isothermal sound speed, Ω K ≡ (GM/r 3 ) 1/2 is the Keplerian angular velocity of the disk, and M is the mass of the central object. The function g(z) accounts for the absence of turbulence in the surface layer of the disk (Bisnovatyi-Kogan & Lovelace 2007;Rothstein & Lovelace 2008). In the body of the disk g = 1, whereas at the surface of the disk, at say z S , g tends over a short distance to a very small value ∼ 10 -8 , effectively zero, which is the ratio of the Spitzer diffusivity of the disk’s surface layer to the turbulent diffusivity of the body of the disk. At the disk’s surface the density is much smaller than its midplane value.
We consider stationary solutions of equations ( 1) and ( 2) for a weak large-scale magnetic field. These can be greatly simplified for thin disks where the disk half-thickness, of the order of h ≡ c s0 /Ω K , is much less than r. Thus we have the small parameter
It is useful in the following to use the dimensionless height ζ ≡ z/h. The midplane plasma beta is taken to be
where
is the midplane Alfvén velocity. Note that the conventional definition of beta is 2β. The rough condition for the MRI instability and the associated turbulence in the disk is β ∼ > 1 (Balbus & Hawley 1998) and this is assumed here.
The three magnetic field components are assumed to be of comparable magnitude on the disk’s surface, but B r = 0 = B φ on the midplane. On the other hand the axial magnetic field changes by only a small almount going from the midplane to the surface, ∆B z ∼ εB r ≪ B z (from ∇ • B = 0) so that B z ≈ const inside the disk. As a consequence, the ∂B j /∂r terms in the magnetic force in equation ( 1) can all be dropped in favor of the ∂B j /∂z t
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