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Register a new userThe Parker Instability under a Linear Gravity
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Authors
Jongsoo Kim
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A linear stability analysis has been done to a magnetized disk under a linear gravity. We have reduced the linearized perturbation equations to a second-order differential equation which resembles the Schr{o}dinger equation with the potential of a harmonic oscillator. Depending on the signs of energy and potential terms, eigensolutions can be classified into ``continuum and ``discrete families. When magnetic field is ignored, the continuum family is identified as the convective mode, while the discrete family as acoustic-gravity waves. If the effective adiabatic index $gamma$ is less than unity, the former develops into the convective instability. When a magnetic field is included, the continuum and discrete families further branch into several solutions with different characters. The continuum family is divided into two modes: one is the original Parker mode, which is a slow MHD mode modulated by the gravity, and the other is a stable Alfven mode. The Parker modes can be either stable or unstable depending on $gamma$. When $gamma$ is smaller than a critical value $gamma_{cr}$, the Parker mode becomes unstable. The discrete family is divided into three modes: a stable fast MHD mode modulated by the gravity, a stable slow MHD mode modulated by the gravity, and an unstable mode which is also attributed to a slow MHD mode. The unstable discrete mode does not always exist. Even though the unstable discrete mode exists, the Parker mode dominates it if the Parker mode is unstable. However, if $gamma ge gamma_{cr}$, the discrete mode could be the only unstable one. When $gamma$ is equal $gamma_{cr}$, the minimum growth time of the unstable discrete mode is $1.3 times 10^8$ years with a corresponding length scale of 2.4 kpc. It is suggestive that the corrugatory features seen in the Galaxy and external galaxies are related to the unstable discrete mode.
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We examine the evolution of the Parker instability in galactic disks using 3D numerical simulations. We consider a local Cartesian box section of a galactic disk, where gas, magnetic fields and cosmic rays are all initially in a magnetohydrostatic equilibrium. This is done for different choices of initial cosmic ray density and magnetic field. The growth rates and characteristic scales obtained from the models, as well as their dependences on the density of cosmic rays and magnetic fields, are in broad agreement with previous (linearized, ideal) analytical work. However, this non-ideal instability develops a multi-modal 3D structure, which cannot be quantitatively predicted from the earlier linearized studies. This 3D signature of the instability will be of importance in interpreting observations. As a preliminary step towards such interpretations, we calculate synthetic polarized intensity and Faraday rotation measure maps, and the associated structure functions of the latter, from our simulations; these suggest that the correlation scales inferred from rotation measure maps are a possible probe for the cosmic ray content of a given galaxy. Our calculations highlight the importance of cosmic rays in these measures, making them an essential ingredient of realistic models of the interstellar medium.
The Parker instability, which has been considered as a process governing the structure of the interstellar medium, is induced by the buoyancy of magnetic field and cosmic rays. In previous studies, while the magnetic field has been fully incorporated in the context of isothermal magnetohydrodynamics, cosmic rays have been normally treated with the simplifying assumption of infinite diffusion along magnetic field lines but no diffusion across them. The cosmic ray diffusion is, however, finite. In this work, we take into account fully the diffusion process of cosmic rays in a linear stability analysis of the Parker instability. Cosmic rays are described with the diffusion-convection equation. With realistic values of cosmic ray diffusion coefficients expected in the interstellar medium, we show that the result of previous studies with the simplifying assumption on cosmic ray diffusion applies well. Finiteness of parallel diffusion decreases the growth rate of the Parker instability, while the relatively smaller perpendicular diffusion has no significant effect. We discuss the implication of our result on the role of the Parker instability in the interstellar medium.
Hydromagnetic stresses in accretion discs have been the subject of intense theoretical research over the past one and a half decades. Most of the disc simulations have assumed a small initial magnetic field and studied the turbulence that arises from the magnetorotational instability. However, gaseous discs in galactic nuclei and in some binary systems are likely to have significant initial magnetisation. Motivated by this, we performed ideal magnetohydrodynamic simulations of strongly magnetised, vertically stratified discs in a Keplerian potential. Our initial equilibrium configuration, which has an azimuthal magnetic field in equipartion with thermal pressure, is unstable to the Parker instability. This leads to the expelling of magnetic field arcs, anchored in the midplane of the disc, to around five scale heights from the midplane. Transition to turbulence happens primarily through magnetorotational instability in the resulting vertical fields, although magnetorotational shear instability in the unperturbed azimuthal field plays a significant role as well, especially in the midplane where buoyancy is weak. High magnetic and hydrodynamical stresses arise, yielding an effective $alpha$-value of around 0.1 in our highest resolution run. Azimuthal magnetic field expelled by magnetic buoyancy from the disc is continuously replenished by the stretching of a radial field created as gas parcels slide in the linear gravity field along inclined magnetic field lines. This dynamo process, where the bending of field lines by the Parker instability leads to re-creation of the azimuthal field, implies that highly magnetised discs are astrophysically viable and that they have high accretion rates.
Here we present three-dimensional MHD models for the Parker instability in a thick magnetized disk, including the presence of a spiral arm. The $B$-field is assumed parallel to the arm, and the model results are applied to the optical segment of the Carina-Sagittarius arm. The characteristic features of the undular and interchange modes are clearly apparent in the simulations. The undular mode creates large gas concentrations distributed along the arm. This results in a clear arm/inter-arm difference: the instability triggers the formation of large interstellar clouds inside the arms, but generates only small structures with slight density enhancements in the inter-arm regions. The resulting clouds are distributed in an antisymmetric way with respect to the midplane, creating an azimuthal corrugation along the arm. For conditions similar to those of the optical segment of the Carina-Sagittarius arm, it has a wavelength of about 2.4 kpc. This structuring can explain the origin of both HI superclouds and the azimuthal corrugations in spiral arms. The wavelength matches the corrugation length derived with the young stellar groups located in the optical segment of the Carina-Sagittarius arm. Keywords: Galaxy: kinematics and dynamics -- Galaxy: structure -- Instabilities -- ISM: clouds -- ISM: magnetic fields -- ISM: structure -- MHD
We study the question of stability of the ground state of a scalar theory which is a generalization of the phi^3 theory and has some similarity to gravity with a cosmological constant. We show that the ground state of the theory at zero temperature becomes unstable above a certain critical temperature, which is evaluated in closed form at high temperature.
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