Global model of differential rotation in the Sun

The isorotation contours of the solar convective zone (SCZ) show three distinct morphologies, corresponding to two boundary layers (inner and outer), and the bulk of the interior. Previous work has shown that the thermal wind equation together with informal arguments on the nature of convection in a rotating fluid could be used to deduce the shape of the isorotation surfaces in the bulk of the SCZ with great fidelity, and that the tachocline contours could also be described by relatively simple phenomenology. In this paper, we show that the form of these surfaces can be understood more broadly as a mathematical consequence of the thermal wind equation and a narrow convective shell. The analysis does not yield the angular velocity function directly, an additional surface boundary condition is required. But much can already be deduced without constructing the entire rotation profile. The mathematics may be combined with dynamical arguments put forth in previous works to the mutual benefit of each. An important element of our approach is to regard the constant angular velocity surfaces as an independent coordinate variable for what is termed the "residual entropy," a quantity that plays a key role in the equation of thermal wind balance. The difference between the dynamics of the bulk of the SCZ and the tachocline is due to a different functional form of the residual entropy in each region. We develop a unified theory for the rotational behavior of both the SCZ and the tachocline, using the solutions for the characteristics of the thermal wind equation. These characteristics are identical to the isorotation contours in the bulk of the SCZ, but the two deviate in the tachocline. The outer layer may be treated, at least descriptively, by similar mathematical techniques, but this region probably does not obey thermal wind balance.Comment: 26 pages, 7 figures, accepted to MNRA

Ambipolar Diffusion in the Magnetorotational Instability

The effects of ambipolar diffusion on the linear stability of weakly ionised accretion discs are examined. Earlier work on this topic has focused on axial magnetic fields and perturbation wavenumbers. We consider here more general field and wavenumber geometries, and find that qualitatively new results are obtained. Provided a radial wavenumber and azimuthal field are present along with their axial counterparts, ambipolar diffusion will always be destabilising, with unstable local modes appearing at well-defined wavenumber bands. The wavenumber corresponding to the maximum growth rate need not, in general, lie along the vertical axis. Growth rates become small relative to the local angular velocity when the ion-neutral collision time exceeds the orbital time. In common with Hall electromotive forces, ambipolar diffusion destabilises both positive and negative angular velocity gradients. In at least some cases, therefore, uniformly rotating molecular cloud cores may reflect the marginally stable state of the ambipolar magnetorotational instability.Comment: Submitted to MN, 6 pages, 3 figs, MN style file v2.

Finite Larmor Radius Effects on Weakly Magnetized, Dilute Plasmas

We investigate the stability properties of a hot, dilute and differentially rotating weakly magnetized plasma which is believed to be found in the interstellar medium of galaxies and protogalaxies and in the low-density accretion flows around some giant black holes like the one in the Galactic center. In the linear MHD regime, we consider the combined effects of gyroviscosity and parallel viscosity on the magnetorotational instability. The helical magnetic field is considered in the investigation. We show that the gyroviscous effect and the pitch angles cause a powerful gyroviscous instability. Furthermore, in most of the cases, plasma with the above mentioned properties is unstable and the growth rates of the unstable modes are larger than that of the magnetorotational instability.Comment: 7 pages, 4 figures. Accepted for publication in MNRA

A Simple Model for Solar Isorotational Contours

The solar convective zone, or SCZ, is nearly adiabatic and marginally convectively unstable. But the SCZ is also in a state of differential rotation, and its dynamical stability properties are those of a weakly magnetized gas. This renders it far more prone to rapidly growing rotational baroclinic instabilities than a hydrodynamical system would be. These instabilities should be treated on the same footing as convective instabilites. If isentropic and isorotational surfaces coincide in the SCZ, the gas is marginally (un)stable to {\em both} convective and rotational disturbances. This is a plausible resolution for the instabilities associated with these more general rotating convective systems. This motivates an analysis of the thermal wind equation in which isentropes and isorotational surfaces are identical. The characteristics of this partial differential equation correspond to isorotation contours, and their form may be deduced even without precise knowledge of how the entropy and rotation are functionally related. Although the exact solution of the global SCZ problem in principle requires this knowledge, even the simplest models produce striking results in broad agreement with helioseismology data. This includes horizontal (i.e. quasi-spherical) isorotational contours at the poles, axial contours at the equator, and approximately radial contours at midlatitudes. The theory does not apply directly to the tachocline, where a simple thermal wind balance is not expected to be valid. The work presented here is subject to tests of self-consistency, among them the prediction that there should be good agreement between isentropes and isorotational contours in sufficiently well-resolved large scale numerical MHD simulations.Comment: Final version: 21 pages, 4 figures, to appear in MNRAS; thorough revision, typos and minor errors corrected, expanded development and reordering of the material. Conclusions unchanged from origina

The stability of stratified, rotating systems and the generation of vorticity in the Sun

We examine the linear behavior of three-dimensional Lagrangian displacements in a stratified, shearing background. The isentropic and iso-rotation surfaces of the equilibrium flow are assumed to be axisymmetric, but otherwise fully two-dimensional. Three-dimensional magnetic fields are included in the perturbation equations; however the equilibrium is assumed to be well-described by purely hydrodynamic forces. The model, in principle very general, is used to study the behavior of fluid displacements in an environment resembling the solar convection zone. Some very suggestive results emerge. All but high-latitude displacements align themselves with the observed surfaces of constant angular velocity. The tendency for the angular velocity to remain constant with depth in the bulk of the convective zone, together with other critical features of the rotation profile, emerge from little more than a visual inspection of the governing equation. In the absence of a background axial angular velocity gradient, displacements exhibit no poleward bias, suggesting that solar convection "plays-off" of prexisting shear rather than creates it. We argue that baroclinic vorticity of precisely the right order is generated at the radiative/convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces. If so, many features of the Sun's internal rotation become more clear, including: i) the general appearance of the tachocline; ii) the extension of differential rotation well into the radiative zone; iii) the abrupt change of morphology of convective zone isorotation surfaces; and iv) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions.Comment: 30 pages, 2 figures. Accepted for publication in MNRA

Thermal instability in rotating galactic coronae

The thermal stability of rotating, stratified, unmagnetized atmospheres is studied by means of linear-perturbation analysis, finding stability, overstability or instability, depending on the properties of the gas distribution, but also on the nature of the perturbations. In the relevant case of distributions with outward-increasing specific entropy and angular momentum, axisymmetric perturbations grow exponentially, unless their wavelength is short enough that they are damped by thermal conduction; non-axisymmetric perturbations typically undergo overstable oscillations in the limit of zero conductivity, but are effectively stabilized by thermal conduction, provided rotation is differential. To the extent that the studied models are representative of the poorly constrained hot atmospheres of disc galaxies, these results imply that blob-like, cool overdensities are unlikely to grow in galactic coronae, suggesting an external origin for the high-velocity clouds of the Milky Way.Comment: 18 pages, 5 figures. Accepted for publication in MNRA

Thermal stability of a weakly magnetized rotating plasma

The thermal stability of a weakly magnetized, rotating, stratified, optically thin plasma is studied by means of linear-perturbation analysis. We derive dispersion relations and criteria for stability against axisymmetric perturbations that generalize previous results on either non-rotating or unmagnetized fluids. The implications for the hot atmospheres of galaxies and galaxy clusters are discussed.Comment: 16 pages, 3 figures, MNRAS accepted. New figures and corrected equations with respect to previous version. Results unchange

The effect of the tachocline on differential rotation in the Sun

In this paper, we present a model for the effects of the tachocline on the differential rotation in the solar convection zone. The mathematical technique relies on the assumption that entropy is nearly constant ("well-mixed") in isorotation surfaces both outside and within the tachocline. The resulting solutions exhibit nontrivial features that strikingly resemble the true tachocline isorotation contours in unexpected detail. This strengthens the mathematical premises of the theory. The observed rotation pattern in the tachocline shows strong quadrupolar structure, an important feature that is explicitly used in constructing our solutions. The tachocline is treated locally as an interior boundary layer of small but finite thickness, and an explicit global solution is then constructed. A dynamical link can thus be established between the internal jump in the angular velocity at the tachocline and the spread of angular velocities observed near the solar surface. In general, our results suggest that the bulk of the solar convection zone is in thermal wind balance, and that simple quadrupolar stresses, local in radius, mediate the tachocline transition from differential rotation to uniform rotation in the radiative interior.Comment: 20 Pages, 4 figures, to appear in MNRA

The general relativistic thin disc evolution equation

In the classical theory of thin disc accretion discs, the constraints of mass and angular momentum conservation lead to a diffusion-like equation for the turbulent evolution of the surface density. Here, we revisit this problem, extending the Newtonian analysis to the regime of Kerr geometry relevant to black holes. A diffusion-like equation once again emerges, but now with a singularity at the radius at which the effective angular momentum gradient passes through zero. The equation may be analysed using a combination of WKB, local techniques, and matched asymptotic expansions. It is shown that imposing the boundary condition of a vanishing stress tensor (more precisely the radial-azimuthal component thereof) allows smooth stable modes to exist external to the angular momentum singularity, the innermost stable circular orbit, while smoothly vanishing inside this location. The extension of the disc diffusion equation to the domain of general relativity introduces a new tool for numerical and phenomenolgical studies of accretion discs, and may prove to be a useful technique for understanding black hole X-ray transients.Comment: 7 Pages, 1 figure. Accepted for publication in MNRAS. Revised version corrects minor typos in equations (64) and (66) of original, otherwise unaltere