Magnetohydrodynamic Instabilities in Shearing, Rotating, Stratified Winds and Disks

We investigate shear and buoyancy instabilities in radially stratified, magnetized, cylindrical flows, for application to magnetocentrifugally driven winds—such as those from protostars—and 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 instabilities. Using four different 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 fields are predominantly toroidal, as in protostellar winds far from their source, we find the system is susceptible to growth of five different kinds of perturbations: axisymmetric fundamental and toroidal resonance modes, axisymmetric and nonaxisymmetric toroidal buoyancy modes, and nonaxisymmetric magnetorotational modes. Winds having a sufficiently steep field gradient (d ln B/d ln R < -0.75 for a purely toroidal-field 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 equilibrium has nonzero magnetic forces. For low-temperature toroidal-B winds, both axisymmetric and nonaxisymmetric magnetorotational instabilities have very low growth rates. The stabilization of buoyancy 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 fields are predominantly poloidal, as may occur in protostellar winds close to their source or in astrophysical disks, we find 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 magnetic field 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. Previous 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 "coherent wavelet" technique (a WKB temporal approximation) and derive closed-form analytic expressions for instantaneous instability criteria, growth rates, and net amplification factors for generalized nonaxisymmetric magnetorotational instabilities in compressible flows with both poloidal and toroidal fields. We confirm that these are in excellent agreement with the results of shearing-sheet temporal integrations and that "locally axisymmetric" perturbations have the largest amplifications only provided ( ∙A)/Ω ≲ 1.