Magnetohydrodynamic Instabilities in Shearing, Rotating, Stratified Winds and Disks

Abstract

We investigate shear and buoyancy instabilities in radially strati—ed, magnetized, cylindrical —ows, for application to magnetocentrifugally driven windssuch as those from protostarsand to magnetized accretion disks. Our motivation is to characterize the susceptibility of cold MHD disk winds to growing internal perturbations and to understand the relation of wind instabilities to known accretion disk insta- bilities. Using four diUerent linear analysis techniques, we identify and study nine principal types of unstable or overstable disturbances, providing numerical and analytic solutions for growth rates for a wide range of parameters. When magnetic —elds are predominantly toroidal, as in protostellar winds far from their source, we —nd the system is susceptible to growth of —ve diUerent kinds of perturbations: axisymmetric fundamental and toroidal resonance modes, axisymmetric and nonaxisymmetric toroidal buoyancy modes, and nonaxisymmetric magnetorotational modes. Winds having a sufficiently steep —eld gradient (d ln B/d ln R (0.75 for a purely toroidal-—eld case) are globally unstable to the long- wavelength fundamental mode concentrated at small radii; these promote the establishment of narrow dense jets in the centers of wider winds. Long-wavelength outer-wind modes are all stable for power-law wind equilibria. The toroidal buoyancy instabilities promote small-scale radial mixing provided the equi- librium has nonzero magnetic forces. For low-temperature toroidal-B winds, both axisymmetric and nonaxisymmetric magnetorotational instabilities have very low growth rates. The stabilization of buoy- ancy instabilities by shear and of magnetorotational instabilities by compressibility may be important in allowing cold MHD winds to propagate over vast distances in space. When magnetic —elds are predomi- nantly poloidal, as may occur in protostellar winds close to their source or in astrophysical disks, we —nd the system is susceptible to four additional growing modes: axisymmetric magnetorotational (Balbus- Hawley), axisymmetric poloidal buoyancy, nonaxisymmetric geometric buoyancy, and poloidal resonance modes. The well-known axisymmetric Balbus-Hawley mode has the fastest growth rate. When the mag- netic —eld is nonuniform, the axisymmetric poloidal buoyancy mode promotes radial mixing on small scales. The geometric poloidal buoyancy mode requires high m, thus is readily stabilized by shear. Pre- vious work on magnetorotational instabilities has concentrated on near-incompressible systems (accretion disks or stellar interiors). We extend this analysis to allow for compressibility (important in winds). We introduce a ii coherent wavelet ˇˇ technique (a WKB temporal approximation) and derive closed-form analytic expressions for instantaneous instability criteria, growth rates, and net ampli—cation factors for generalized nonaxisymmetric magnetorotational instabilities in compressible —ows with both poloidal and toroidal —elds. We con—rm that these are in excellent agreement with the results of shearing-sheet temporal integrations and that ii locally axisymmetric ˇˇ perturbations have the largest ampli—cations only provided (k ˘? A )/) ( 1. Subject headings: accretion, accretion disksISM: jets and out—ows ¨ ISM: kinematics and dynamicsISM: magnetic —eldsMHD ¨ stars: premain-sequence