A Magnetic α ω Dynamo in Active Galactic Nuclei Disks - NMT Physics

– 21 –rotating frame with the origin at the point r = r 0 such that the x-axis is directed radially outward,the y-axis is directed in the positive azimuthal direction, and the z-axis is perpendicular to the diskplane. Then, the effective gravitational and centrifugal potential in the Keplerian rotating framein the neighborhood of the point r = r 0 is∆Φ = GM2r03 (z 2 − 3x 2 ). (21)The thermal energy of the hot column of gas is a fraction of the loss of kinetic energy of the stardue to the hydrodynamic collision with the disk. This latter energy loss during one passage isF drag 2H = 2HπR 2 ⊙ρv 2 ∗ = πR 2 ⊙Σv 2 ∗.Without a hydrodynamic simulation in 3-dimensions an accurate description is missing. Neverthelessit is sufficient to approximate the solution as that fraction of the disk matter that hasan internal energy density, ǫ shk greater than that of the ambient disk by that factor such that itwill rise to a height, z, determined by its potential energy, or ∆Φ = GMz22r 3 (equation (21)). Since∆S/S o = ǫ shk /ǫ o , where ǫ o ≃ ∆Φ(H) = GMH22r 3 , then in order for a plume to rise above the diskmid-plane to a height, l,( lH) 2≃(ǫshk)≃ǫ o( ) 2 ( ) 2 vK R⊙( r) ( ) 2 2 R⊙≃ . (22)c s R shk H R shkWe are concerned with plumes that rise well above the disk so that they can expand horizontallyby a factor several times the plume’s original radius. In this case the moment of inertia of theplume about its own axis will be increased by several times before falling back to the disk. Thiscauses the plume to reduce its own rotation rate relative to the frame of the disk, that is to untwistrelative to that frame. For this expansion to take place, the plume must rise roughly ∼ 2H abovethe disk, or l ≃ 3H. At this height the pressure of the hydrostatic isothermal atmosphere with thedensity profile as ∝ exp(−z 2 /H 2 ) becomes negligible compared to that of the plume, and so thehot gas of the plume can expand both vertically and horizontally as a free expansion. With this lwe getR shk≃ 1 rR ⊙ 3 H . (23)Using expression (8) for H/r at r ≤ r ab we obtainR shk ≃ 0.24H( αss) ( ) (2/21−26/21lE(ǫ) 26/21−19/21 r M 8 1 −0.01 0.1 0.1 r ab√ ) −23rg, (24)Thus a plume, starting from a size, R shk < H will expand to a size ≃ 2H both vertically andhorizontally, thus producing a near spherical bubble with radius r p = H above the disk. Post shockexpansion will increase the estimate of R shk somewhat. For simplicity we will use R shk = H/2 forestimates of the toroidal flux entrained in the plumes in paper II. This is our standard plume.r