System of Equations which leads to MRI: This system is linearized about an initial state where the fluid is in Keplerian rotation and B is vertical. The terms in {} apply only in this case. The equations are integrated vertically from the midplane to the surface. Jets: The system is integrated vertically from the midplane until the position of the sonic point and values of the variables there can be estimated. Then the solution is integrated backwards to a fitting point. The process is iterated until the full solution converges. This solution is then matched onto a global (self-similar) wind solution, imposing the Alfvén critical point constraint. Magnetic fields provide mechanisms for removing angular momentum in protoplanetary disks. These include MHD turbulence involving a disordered, small- scale field and hydromagnetic outflows launched by large-scale fields. Results derived from recent HST observations support the magneto-centrifugal model of jet launching and indicate that, at least at small radii (R  1 AU), a large fraction of the excess angular momentum can be carried away by the outflow [8]. Previous investigations have considered the above two mechanisms separately. We have developed a quantitative scheme, based on numerical simulations and analytic solutions, for treating `hybrid’ disks in which both radial and vertical angular momentum transport may occur at the same radial location. Because of the low ionization fraction of the fluid, the magnetic coupling is weak and depends strongly on the conductivity and its spatial variation. Our models take account of the vertical structure of the disk and of a realistic conductivity. The contributions of ambipolar diffusion, the Hall conductivity and Ohmic resistivity are all included.AbstractParameters Normalized midplane radial speed (-V ro /c s ) The following parameters are also relevant for the wind solutions: Field-aligned conductivity a 0 Ratio of the Alfven to the sound speed at the midplane (v A0 /c s ) Hall conductivity Magnetic coupling at the midplane Pedersen conductivity Further Progress -Complete analysis of wind solutions using a realistic ionization profile. -Match local wind solutions onto self-similar global wind solutions [2, 7, 3]. -Model the fractions of angular momentum transported via the MRI and outflows, respectively, as a function of position and magnetic field distribution in protoplanetary disks. -Investigate the implications of the disk structure to planet formation and migration. Magnetorotationalinstability (1) Structure of the perturbations (2) Growth Rates Left panel: Perturbations grow for B < 8 G (1 AU), B < 800 mG (5 AU) and B < 250 mG (10 AU). For a range of B, max is of order the ideal-MHD rate (0.75  Right Panel:Hall conductivity modifies the growth rate of global unstable modes at 1 AU for all magnetic field strengths that support MRI [5]. Left panel: Growth rate of the most unstable modes of the MRI for R = 1, 5 and 10 AU as a function of the strength of the magnetic field for the minimum-solar nebula disk and including the ionization provided by cosmic rays, X-rays and radioactive decay. Right panel: Growth rates of the fastest growing modes at 1 AU as a function of the strength of the magnetic field for different configurations of the conductivity tensor Jets & Outflows Angular Momentum Transport in Protoplanetary Disks Raquel Salmeron (1), Arieh Königl (1) & Mark Wardle (2) (1) Dept. Astronomy & Astrophysics, University of Chicago (2) Dept. Physics, Macquarie University hydrostatic transition wind B z=0 zszs zbzb zhzh Representative field line and poloidal velocity of the fluid (arrows). Z h denotes the disk scale height, z b is the base of the wind and z s is the sonic point. (1) Illustrative vertical disk structure Adapted from [6] (3) Realistic solution at 1 AU Representative wind solution as a function of height for R = 1 AU, calculated with a realistic cosmic ray ionization profile. Left panels: Density profile and magnetic field components (top panel) and velocity components (bottom). Quantities are normalized by their midplane values. Right panels: Conductivity tensor components (top panel) and magnetic coupling and normalized field strength (bottom panel). To obtain this solution we guess the surface density (  ) and calculate the ionization profile and wind structure.  is then iterated until the solution converges. The sonic point lies at z/H = 2.2 and Hall conductivity dominates for z/H  0.4. Note that the magnetic coupling (  ) increases with the ionization rate at low z and then drops with the fluid density. References: References: [1] Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214. [2] Blandford R. D., Payne D.G., 1982, MNRAS, 199, 883. [3] Li Z. Y., 1995, ApJ, 444, 848. [4] Ogilvie G., Livio M., 2001, ApJ 553, 158. [5] Salmeron R. & Wardle M., 2005, MNRAS, 361, 45. [6] Sano T., et al., 2004, ApJ, 605, 321. [7] Wardle M. & Königl A., 1993, ApJ, 410, 218. [8] Woitas et. al., 2005, A&A, 432, 149. Top panel: Radial (solid lines) and azimuthal (dashed lines) field components for solutions in the ambipolar diffusion limit and different field strengths. a o = 1 (magenta), 0.8 (blue), 0.6 (green), 0.5 (red) and 0.47 (black). Bottom panel: B r /B o for weak-field solutions in the ambipolar diffusion limit. The case where only vertical angular momentum transport is included is displayed in blue while the red line shows a solution that incorporates radial angular momentum transport. Top panel: As |B| decreases the field lines bend more. For weak enough fields, dB r /dz may become 0 within the disk. Left panels: Full conductivity modes have higher wavenumber, and grow closer to the midplane, than modes in the ambipolar diffusion limit. Right panels: When dust grains are present, both the wavenumber and growth rate of the perturbations are reduced and the dead zone is more extended. Structure of the most unstable modes of the MRI for the minimum-mass solar nebula model as a function of the magnetic field strength (top right corner of each panel). The maximum growth rate ( max ) is shown in bottom right corners. Solid lines display  B r and dashed ones correspond to  B . Left panel: At R = 1 AU for different configurations of the conductivity tensor. Right panel: R = 10 AU. The left column shows the case where dust grains have settled, while the right one displays results when 0.1  m grains are present. Combined solutions: Using expressions from [6], the space and volume-averaged Maxwell stress > is written in terms of B o and a 2. The term >/r is added to the angular momentum equation in the region where the field is deemed weak enough for MRI to operate (see below). The term >/r is added to the angular momentum equation in the region where 2  a 2 1 (the wind region) B is strong enough to suppress MRI modes. (2) Solutions in parameter space Left: When Hall conductivity is important, the hydrostatic region is more extended and the sonic point lies higher above z=0 than in the ambipolar diffusion (AD) or resistive limits. Middle: The density profile of the solutions is steeper than in the pure tidal-squeezing case (the disk is magnetically confined). In the AD limit, |B| is smaller and the profile is less compressed. Right: The density  s at the sonic point (a measure of the wind’s mass flux) is reduced when the Hall term is present. Structure of representative wind solutions for different configurations of the conductivity tensor. Green lines show the ambipolar diffusion limit and red lines the resistive limit. Blue lines correspond to the case where both components are important. Left panel: Normalized velocity components. Middle panel: Density and magnetic field components profiles normalised by the midplane values. Right panel: Density at the sonic point, a measure of the wind’s mass flux. In the hydrostatic layer, magnetic field lines are radially bent and sheared. The field takes angular momentum from the matter. In the transition layer, field lines are nearly straight as magnetic energy dominates. They overtake the fluid at z b and accelerate matter centrifugally. The zigzag line shows where radial angular momentum transport could occur, if the local B is small enough. (4) Wind – MRI solutions > ~ 0.5 >/8  > ~ 28 > These expressions, lead to > ~ 14Y(B z,o 2 /8  ) (a) Prescription Typical weak-field solution, in the hydrostatic limit, of the type discussed in [4]. To the left of the dashed line, 2  a 2 < 1 and B r bends more than once. (b) Illustrative solutions MRI-stable 2  a 2 > 1 MRIunstable 2  a 2 < 1 MRIunstable [6], where >/B z,o 2 ≡ Y Y ≡ Y(a) ≈ ; B z,o ≡ B z (t=0) = B o MRI stable (tidal squeezing)  =1