Three-dimensional simulations of shear instabilities in magnetized flows

Shear mixing is believed to be the main mechanism to provide extra mixing in stellar interiors. We present results of three-dimensional (3D) simulations of the magnetohydrodynamic Kelvin–Helmholtz instability in a stratified shear layer. The magnetic field is taken to be uniform and parallel to the shear flow. We describe the evolution of the fluid flow and the magnetic field for a range of initial conditions. In particular, we investigate how the mixing rate of the fluid depends on the Richardson number and the magnetic field strength. It is found that the magnetic field can enhance as well as suppress mixing. Moreover, we have performed two-dimensional (2D) simulations and discuss some interesting differences between the 2D and 3D results.     

On a physical mechanism for extra mixing in globular cluster red giants

For the first time we propose a real physical mechanism for 'extra mixing' in red giants that can quantitatively interpret all the known star-to-star abundance variations in globular clusters. This is Zahn's mechanism. It considers extra mixing in a radiative zone of a rotating star as a result of the joint operation of meridional circulation and turbulent diffusion. It is shown that the only free parameter, the angular velocity at the base of the convective envelope, can be so adjusted as to fit the observed abundance correlations without leading to a conflict with available data on rotation velocities of blue horizontal branch stars in the same cluster. There are two critical assumptions in our model, that the top of the radiative zone is not in synchronous rotation with the stellar surface but rotates significantly faster and that the criterion for shear instability takes a particular form. These will eventually be tested by three-dimensional hydrodynamical simulations.     

Subcritical dynamos in shear flows

Identifying generic physical mechanisms responsible for the generation of magnetic fields and turbulence in differentially rotating flows is fundamental to understand the dynamics of astrophysical objects such as accretion disks and stars. In this paper, we discuss the concept of subcritical dynamo action and its hydrodynamic analogue exemplified by the process of nonlinear transition to turbulence in non‐rotating wall‐bounded shear flows. To illustrate this idea, we describe some recent results on nonlinear hydrodynamic transition to turbulence and nonlinear dynamo action in rotating shear flows pertaining to the problem of turbulent angular momentum transport in accretion disks. We argue that this concept is very generic and should be applicable to many astrophysical problems involving a shear flow and non‐axisymmetric instabilities of shearinduced axisymmetric toroidal velocity or magnetic fields, such as Kelvin‐Helmholtz, magnetorotational, Tayler or global magnetoshear instabilities. In the light of several recent numerical results, we finally suggest that, similarly to a standard linear instability, subcritical MHD dynamo processes in high‐Reynolds number shear flows could act as a large‐scale driving mechanism of turbulent flows that would in turn generate an independent small‐scale dynamo. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Triggered Star Formation in Rotating Disks

The gravitational instability of expanding shells triggering the formation of clouds and stars is analyzed. Disks with different scale-heights, ambient and shell velocity dispersions, mid-plane densities, rotation rates and shear rates are explored with three dimensional numerical simulations in the thin shell approximation. Three conditions for the shell collapse are specified: the first is that it happens before a significant blow-out, the second requires that the shell collapses before it is distorted by Coriolis forces and shear, and the third requires that the internal pressure in the accumulated gas is small and the fragmentation is achieved within the expansion time. The gas-rich and slowly rotating galaxies are the best sites of the triggered star formation, concluding that its importance has been much larger at the times of galaxy formation compared to the present epoch. This revised version was published online in September 2006 with corrections to the Cover Date.     

Meridional circulation in the radiation zones of rotating stars: Origins,behaviors and consequences on stellar evolution

Stellar radiation zones are the seat of meridional currents. This circulation has a strong impact on the transport of angular momentum and the mixing of chemicals that modify the evolution of stars. First, we recall in details the dynamical processes that are taking place in differentially rotating stellar radiation zones and the assumptions which are adopted for their modelling in stellar evolution. Then, we present our new results of numerical simulations which allow us to follow in 2D the secular hydrodynamics of rotating stars, assuming that anisotropic turbulence enforces a shellular rotation law and taking into account the transport of angular momentum by internal gravity waves. The different behaviors of the meridional circulation in function of the type of stars which is studied are discussed with their physical origin and their consequences on the transport of angular momentum and of chemicals. Finally, we show how this work is leading to a dynamical vision of the evolution of rotating stars from their birth to their death. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Turbulent stresses as a function of shear rate in a local disk model

We present local numerical models of accretion disk turbulence driven by the magnetorotational instability with varying shear rate. The resulting turbulent stresses are compared with predictions of a closure model in which triple correlations are modelled in terms of quadratic correlations. This local model uses five nondimensional parameters to describe the properties of the flow. We attempt to determine these closure parameters for our simulations and find that the model does produce qualitatively correct behaviour. In addition, we present results concerning the shear rate dependency of the magnetic to kinetic energy ratio. We find both the turbulent stress ratio and the total stress to be strongly dependent on the shear rate (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of accretion discs threaded by a strong magnetic field

We study the stability of poloidal magnetic fields anchored in a thin accretion disc. The two-dimensional hydrodynamics in the disc plane is followed by a grid-based numerical simulation including the vertically integrated magnetic forces. The three-dimensional magnetic field outside the disc is calculated in a potential field approximation from the magnetic flux density distribution in the disc. For uniformly rotating discs we confirm numerically the existence of the interchange instability as predicted by Spruit, Stehle & Papaloizou . In agreement with predictions from the shearing sheet model, discs with Keplerian rotation are found to be stabilized by the shear, as long as the contribution of magnetic forces to support against gravity is small. When this support becomes significant, we find a global instability which transports angular momentum outwardly and allows mass to accrete inwardly. The instability takes the form of a m =1 rotating 'crescent', reminiscent of the purely hydrodynamic non-linear instability previously found in pressure-supported discs. A model where the initial surface mass density Σ( r ) and B _z ( r ) decrease with radius as power laws shows transient mass accretion during about six orbital periods, and settles into a state with surface density and field strength decreasing approximately exponentially with radius. We argue that this instability is likely to be the main angular momentum transport mechanism in discs with a poloidal magnetic field sufficiently strong to suppress magnetic turbulence. It may be especially relevant in jet-producing discs.     

Density wave formation in differentially rotating disk galaxies: Hydrodynamic simulation of the linear regime

Most rapidly and differentially rotating disk galaxies, in which the sound speed (thermal velocity dispersion) is smaller than the orbital velocity, display graceful spiral patterns. Yet, over almost 240 yr after their discovery in M51 by Charles Messier, we still do not fully understand how they originate. In this first paper of a series, the dynamical behavior of a rotating galactic disk is examined numerically by a high-order Godunov hydrodynamic code. The code is implemented to simulate a two-dimensional flow driven by an internal Jeans gravitational instability in a nonresonant wave–“fluid” interaction in an infinitesimally thin disk composed of stars or gas clouds. A goal of this work is to explore the local and linear regimes of density wave formation, employed by Lin, Shu, Yuan and many others in connection with the problem of spiral pattern of rotationally supported galaxies, by means of computer-generated models and to compare those numerical results with the generalized fluid-dynamical wave theory. The focus is on a statistical analysis of time-evolution of density wave structures seen in the simulations. The leading role of collective processes in the formation of both the circular and spiral density waves (“heavy sound”) is emphasized. The main new result is that the disk evolution in the initial, quasilinear stage of the instability in our global simulations is fairly well described using the local approximation of the generalized wave theory. Certain applications of the simulation to actual gas-rich spiral galaxies are also explored.     

Transport of angular momentum and chemical species by anisotropic mixing in stellar radiative interiors

Small levels of turbulence can be present in stellar radiative interiors due to, e.g., the instability of rotational shear. In this paper we estimate turbulent transport coefficients for stably stratified rotating stellar radiation zones. Stable stratification induces strong anisotropy with a very small ratio of radial‐to‐horizontal turbulence intensities. Angular momentum is transported mainly due to the correlation between azimuthal and radial turbulent motions induced by the Coriolis force. This non‐diffusive transport known as the Λ‐effect has outward direction in radius and is much more efficient compared to the effect of radial eddy viscosity. Chemical species are transported by small radial diffusion only. This result is confirmed using direct numerical simulations combined with the test‐scalar method. As a consequence of the non‐diffusive transport of angular momentum, the estimated characteristic time of rotational coupling (≲100 Myr) between radiative core and convective envelope in young solar‐type stars is much shorter compared to the time‐scale of Lithium depletion (∼1 Gyr) (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Instability of Zonal Flows in Rotating Spherical Shells: An Application to Jupiter

Measurements from the Galileo probe suggest that the zonal winds are deep rooted. Jupiter's high rotation rate makes it likely that the whole outer molecular H/He layer is involved in these long-lived jet flows. Assuming that the primary flows are geostrophic, and that the banded surface structure stretches right through the molecular H/He layer, we examine the conditions for such flows to be stable. As a first step, the linear stability of some prescribed banded zonal flows in a rotating spherical shell is explored. Incompressibility is assumed for simplicity, and the boundary condition is stress-free. We compare solutions for two aspect ratios, appropriate for the molecular H/He layers of Jupiter and Saturn, and two Ekman numbers (E=10⁻² and E=10⁻⁴). Convective and shear flow instabilities compete in our system. The convective instabilities are of the well-known columnar structure. Shear flow instabilities for the larger Ekman number are similar to the Taylor-Couette instability in rotating annuli. At the lower Ekman number, shear flow instabilities adopt a geostrophic character, assuming the form of rotating columns, similar to the convective instabilities. While the convective instability always sets in outside the tangent cylinder, shear instability can become unstable inside the tangent cylinder. If even a weak zonal flow is present inside the tangent cylinder, the flow is unstable to shear instability. This offers an explanation why the jovian zonal jet structure is much weaker at the higher latitudes that correspond to locations inside the tangent cylinder.     

A numerical study of the r-mode instability of rapidly rotating nascent neutron stars

The first results of numerical analysis of classical r-modes of rapidly rotating compressible stellar models are reported. The full set of linear perturbation equations of rotating stars in Newtonian gravity is solved numerically without the slow rotation approximation. A critical curve of gravitational wave emission induced instability, which restricts the rotational frequencies of hot young neutron stars, is obtained. Taking the standard cooling mechanisms of neutron stars into account, we also show the 'evolutionary curves' along which neutron stars are supposed to evolve as cooling and spinning down proceed. Rotational frequencies of 1.4-M_⊙ stars suffering from this instability decrease to around 100 Hz when the standard cooling mechanism of neutron stars is employed. This result confirms the results of other authors, who adopted the slow rotation approximation.     

Protodiscs around hot magnetic rotator stars

We develop equations and obtain solutions for the structure and evolution of a protodisc region that is initially formed with no radial motion and super-Keplerian rotation speed when wind material from a hot rotating star is channelled towards its equatorial plane by a dipole-type magnetic field. Its temperature is around 10⁷ K because of shock heating and the inflow of wind material causes its equatorial density to increase with time. The centrifugal force and thermal pressure increase relative to the magnetic force and material escapes at its outer edge. The protodisc region of a uniformly rotating star has almost uniform rotation and will shrink radially unless some instability intervenes. In a star with angular velocity increasing along its surface towards the equator, the angular velocity of the protodisc region decreases radially outwards and magnetorotational instability (MRI) can occur within a few hours or days. Viscosity resulting from MRI will readjust the angular velocity distribution of the protodisc material and may assist in the formation of a quasi-steady disc. Thus, the centrifugal breakout found in numerical simulations for uniformly rotating stars does not imply that quasi-steady discs with slow outflow cannot form around magnetic rotator stars with solar-type differential rotation.     

Saturation of the magnetorotational instability at large Elsasser number

The magnetorotational instability is investigated within the shearing box approximation in the large Elsasser number regime. In this regime, which is of fundamental importance to astrophysical accretion disk theory, shear is the dominant source of energy, but the instability itself requires the presence of a weaker vertical magnetic field. Dissipative effects are weaker still but not negligible. The regime explored retains the condition that (viscous and ohmic) dissipative forces do not play a role in the leading order linear instability mechanism. However, they are sufficiently large to permit a nonlinear feedback mechanism whereby the turbulent stresses generated by the MRI act on and modify the local background shear in the angular velocity profile. To date this response has been omitted in shearing box simulations and is captured by a reduced pde model derived here from the global MHD fluid equations using multiscale asymptotic perturbation theory. Results from numerical simulations of the reduced pde model indicate a linear phase of exponential growth followed by a nonlinear adjustment to algebraic growth and decay in the fluctuating quantities. Remarkably, the velocity and magnetic field correlations associated with these algebraic growth and decay laws conspire to achieve saturation of the angular momentum transport. The inclusion of subdominant ohmic dissipation arrests the algebraic growth of the fluctuations on a longer, dissipative time scale. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

3-D non-linear evolution of a magnetic flux tube in a spherical shell: The isentropic case

We present recent 3-D MHD numerical simulations of the non-linear dynamical evolution of magnetic flux tubes in an adiabatically stratified convection zone in spherical geometry, using the anelastic spherical harmonic (ASH) code.We seek to understand the mechanism of emergence of strong toroidal fields from the base of the solar convection zone to the solar surface as active regions. We confirm the results obtained in cartesian geometry that flux tubes that are not twisted split into two counter vortices before reaching the top of the convection zone. Moreover, we find that twisted tubes undergo the poleward-slip instability due to an unbalanced magnetic curvature force which gives the tube a poleward motion both in the non-rotating and in the rotating case. This poleward drift is found to be more pronounced on tubes originally located at high latitudes. Finally, rotation is found to decrease the rise velocity of the flux tubes through the convection zone, especially when the tube is introduced at low latitudes. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Magnetic and vertical shear instabilities in accretion discs

The stability properties of magnetized discs rotating with angular velocity Ω = Ω( s ,  z ), dependent on both the radial and the vertical coordinates s and z , are considered. Such a rotation law is adequate for many astrophysical discs (e.g., galactic and protoplanetary discs, as well as accretion discs in binaries). In general, the angular velocity depends on height, even in thin accretion discs. A linear stability analysis is performed in the Boussinesq approximation, and the dispersion relation is obtained for short-wavelength perturbations. Any dependence of Ω on z can destabilize the flow. This concerns primarily small-scale perturbations for which the stabilizing effect of buoyancy is strongly suppressed due to the energy exchange with the surrounding plasma. For a weak magnetic field, instability of discs is mainly associated with vertical shear, whilst for an intermediate magnetic field the magnetic shear instability, first considered by Chandrasekhar and Velikhov, is more efficient. This instability is caused by the radial shear which is typically much stronger than the vertical shear. Therefore the growth time for the magnetic shear instability is much shorter than for the vertical shear instability. A relatively strong magnetic field can suppress both these instabilities. The vertical shear instability could be the source of turbulence in protoplanetary discs, where the conductivity is low.     

Hydrodynamic simulations of rotating molecular jets

Molecular outflows and the jets which may drive them can be expected to display signatures associated with rotation if they are the channels through which angular momentum is extracted from material accreting on to protostars. Here, we determine some basic signatures of rapidly rotating flows through three-dimensional numerical simulations of hydrodynamic jets with molecular cooling and chemistry. We find that these rotating jets generate a broad advancing interface which is unstable and develops into a large swarm of small bow features. In comparison to precessing jets, there is no stagnation point along the axis. The greater the rotation rate, the greater the instability. On the other hand, velocity signatures are only significant close to the jet inlet since jet expansion rapidly reduces the rotation speed. We present predictions for atomic, H₂ and CO submillimetre images and spectroscopy including velocity channel maps and position–velocity diagrams. We also include simulated images corresponding to Spitzer IRAC band images and CO emission, relevant for APEX and eventual ALMA observations. We conclude that protostellar jets often show signs of slow precession but only a few sources display properties which could indicate jet rotation.     

Nonlinear evolution of Tayler unstable equilibrium states

We investigate the nonlinear evolution of Tayler unstable toroidal fields in a disk geometry, by numerical simulations. We consider non‐rotating and rigid rotating disks, with different radial field profiles. The initial configuration is in equilibrium, which is achieved by a pressure gradient or an external potential force. The nonlinear evolution of the system leads to a stable equilibrium with a current free toroidal field. Only for the fast rotating case we could preserve a ring type structure of the toroidal field. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Is the Parker Instability a Viable Mechanism for the Formation of Giant Molecular Clouds?

The stratified interstellar medium has been known to be subject to the Parker instability, and the instability has been considered as a plausible mechanism for the formation of giant molecular clouds (GMCs). By presenting our recent efforts to understand the instability through linear analyses and numerical simulations, we raise a negative point of view that the Parker instability alone could not form GMCs.     

Influence of Ohmic diffusion on the excitation and dynamics of MRI

In this paper we make an effort to understand the interaction of turbulence generated by the magnetorotational instability (MRI) with turbulence from other sources, such as supernova explosions (SNe) in galactic disks. First we perform a linear stability analysis (LSA) of non‐ideal MRI to derive the limiting value of Ohmic diffusion that is needed to inhibit the growth of the instability for different types of rotation laws. With the help of a simple analytical expression derived under first‐order smoothing approximation (FOSA), an estimate of the limiting turbulence level and hence the turbulent diffusion needed to damp the MRI is derived. Secondly, we perform numerical simulations in local cubes of isothermal nonstratified gas with external forcing of varying strength to see whether the linear result holds for more complex systems. Purely hydrodynamic calculations with forcing, rotation and shear are made for reference purposes, and as expected, non‐zero Reynolds stresses are found. In the magnetohydrodynamic calculations, therefore, the total stresses generated are a sum of the forcing and MRI contributions. To separate these contributions, we perform reference runs with MRI‐stable shear profiles (angular velocity increasing outwards), which suggest that the MRI‐generated stresses indeed become strongly suppressed as function of the forcing. The Maxwell to Reynolds stress ratio is observed to decrease by an order of magnitude as the turbulence level due to external forcing exceeds the predicted limiting value, which we interpret as a sign of MRI suppression. Finally, we apply these results to estimate the limiting radius inside of which the SN activity can suppress the MRI, arriving at a value of 14 kpc (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the asphericity of nova remnants caused by rotating white dwarf envelopes

The effect of rotating white dwarf envelopes in determining the structure of nova shells is examined. This is achieved by numerical hydrodynamic simulations of the flows around a binary star system. In previous studies of remnant formation, this rotation has not been included.
It is found that the structures formed in the flow are more consistent with observations of nova shells than the previous theoretical studies. The shells produced by the nova become more prolate with increasing white dwarf envelope rotation. Hence the rotation of white dwarf envelopes must be included in any future discussion of remnant formation.
A possible method of identifying the dominant process by which mixing of accreted and white dwarf matter takes place is suggested.