Turbulent convection in rapidly rotating spherical shells: A model for equatorial and high latitude jets on Jupiter and Saturn

The origin of zonal jets on the jovian planets has long been a topic of scientific debate. In this paper we show that deep convection in a spherical shell can generate zonal flow comparable to that observed on Jupiter and Saturn, including a broad prograde equatorial jet and multiple alternating jets at higher latitudes. We present fully turbulent, 3D spherical numerical simulations of rapidly rotating convection with different spherical shell geometries. The resulting global flow fields tend to be segregated into three regions (north, equatorial, and south), bounded by the tangent cylinder that circumscribes the inner boundary equator. In all of our simulations a strong prograde equatorial jet forms outside the tangent cylinder, whereas multiple jets form in the northern and southern hemispheres, inside the tangent cylinder. The jet scaling of our numerical models and of Jupiter and Saturn is consistent with the theory of geostrophic turbulence, which we extend to include the effect of spherical shell geometry. Zonal flow in a spherical shell is distinguished from that in a full sphere or a shallow layer by the effect of the tangent cylinder, which marks a reversal in the sign of the planetary β-parameter and a jump in the Rhines length. This jump is manifest in the numerical simulations as a sharp equatorward increase in jet widths—a transition that is also observed on Jupiter and Saturn. The location of this transition gives an estimate of the depth of zonal flow, which seems to be consistent with current models of the jovian and saturnian interiors.     

On the local magnetorotational instability of astrophysical jets

We present the local linear stability analysis of rotating jets confined by a toroidal magnetic field. Under the thin flux tube approximation, we derive the equation of motion for slender magnetic flux tubes. In addition to the terms responsible for the conventional instability of the toroidal magnetic field, a term related to the magnetic buoyancy and a term corresponding to the differential rotation become relevant for the stability properties. We find that the rigid rotation stabilizes while the differential rotational destabilizes the jet in a way similar to the Balbus–Hawley instability. Within the frame of our local analysis, we find that if the azimuthal velocity is of the order of or higher than the Alfvén azimuthal speed, the rigidly rotating part of the jet interior can be completely stabilized, while the strong shearing instability operates in the transition layer between the rotating jet interior and the external medium. This can explain the limb-brightening effect observed in several jets. However, it is still possible to find jet equilibria that are stable all across the jet, even in the presence of differential rotation. We discuss observational consequences of these results.     

Plasma Jet Studies via the Flow Z-Pinch

The ZaP sheared-flow Z-pinch produces high density Z-pinch plasmas that are stable for up to 2000 times the classical instability times. The presence of an embedded radial shear in the axial flow is correlated with the observed stability, and is in agreement with numerical predictions of the stability threshold. The case is made that using a higher-Z working gas will produce supersonic plasma jets, consistent with dimensionless similarity constraints of astrophysical jets. This would allow laboratory testing of some regimes of astrophysical jet theory, computations, and observations.     

Effect of expansion and magnetic field configuration on mass entrainment of jets

We investigate the growth of jet plus entrained mass in simulations of supermagnetosonic cylindrical and expanding jets. The entrained mass spatially grows in three stages: from an initially slow spatial rate to a faster rate and finally at a flatter rate. These stages roughly coincide with the similar rates of expansion in simulated radio intensity maps, and also appear related to the growth of the Kelvin–Helmholtz instability through linear, nonlinear, and saturated regimes. In the supermagnetosonic cylindrical jets, we found that a jet with an embedded primarily toroidal magnetic field is more stable than a jet with a primarily axial magnetic field. Also, pressure-matched expanding jets are more stable and entrain less mass than cylindrical jets with equivalent inlet conditions. We investigate the growth of jet plus entrained mass in simulations of supermagnetosonic cylindrical and expanding jets. The entrained mass spatially grows in three stages: from an initially slow spatial rate to a faster rate and finally at a flatter rate. These stages roughly coincide with the similar rates of expansion in simulated radio intensity maps, and also appear related to the growth of the Kelvin–Helmholtz instability through linear, nonlinear, and saturated regimes. In the supermagnetosonic cylindrical jets, we found that a jet with an embedded primarily toroidal magnetic field is more stable than a jet with a primarily axial magnetic field. Also, pressure-matched expanding jets are more stable and entrain less mass than cylindrical jets with equivalent inlet conditions.     

Zonal jets in rotating convection with mixed mechanical boundary conditions

Large-scale zonal flows, as observed on the giant planets, can be driven by thermal convection in a rapidly rotating spherical shell. Most previous models of convectively-driven zonal flow generation have utilized stress-free mechanical boundary conditions (FBC) for both the inner and the outer surfaces of the convecting layer. Here, using 3D numerical models, we compare the FBC case to the case with a stress free outer boundary and a non-slip inner boundary, which we call the mixed case (MBC). We find significant differences in surface zonal flow profiles produced by the two cases. In low to moderate Rayleigh number FBC cases, the main equatorial jet is flanked by a strong, high-latitude retrograde jets in the northern and southern hemispheres. For the highest Rayleigh number FBC case, the equatorial jet is flanked by strong reversed jets as well as two additional large-scale alternating jets at higher latitudes. The MBC cases feature stronger equatorial jets but, much weaker, small-scale alternating zonal flows are found at higher latitudes. Our high Rayleigh number FBC results best compare with the zonal flow pattern observed on Jupiter, where the equatorial jet is flanked by strong retrograde jets as well as small-scale alternating jets at high latitude. In contrast, the MBC results compare better with the observed flow pattern on Saturn, which is characterized by a dominant prograde equatorial jet and a lack of strong high latitude retrograde flow. This may suggest that the mechanical coupling at the base of the jovian convection zone differs from that on Saturn.     

Interactions between molecular outflows and optical jets

Young stars produce both molecular outflows and, at a later evolutionary stage, well-collimated optical jets. The simplest explanation is that the molecular outflows are driven byobscured optical jets, rather than directly, by a disk wind for example, but the optical jets appear to have too small a momentum flux. Recent statistical studies however show that the molecular flows must be quasi-stationary, which means that the dynamical lifetime is a gross underestimate of the true age. As a consequence much less thrust is required. We present recent observations of RNO 43, which has well-defined optical and molecular outflows lying close to the plane of the sky. Excellent agreement with the observations is obtained with a simple kinematic model for the molecular material, which supposes that it lies in a parabolic shell around the optical jet with the highest velocities at the working surface. Together with our modelling of the NGC2024 outflow, this is very strong evidence that molecular outflows are produced by prompt entrainment of molecular material in a neutral or weakly-ionized jet.     

The flat synchrotron spectra of partially self-absorbed jets revisited

Flat radio spectra with large brightness temperatures at the core of active galactic nuclei and X-ray binaries are usually interpreted as the partially self-absorbed bases of jet flows emitting synchrotron radiation. Here we extend previous models of jets propagating at large angles to our line of sight to self-consistently include the effects of energy losses of the relativistic electrons due to the synchrotron process itself and the adiabatic expansion of the jet flow. We also take into account energy gains through self-absorption. Two model classes are presented. The ballistic jet flows, with the jet material travelling along straight trajectories, and adiabatic jets. Despite the energy losses, both scenarios can result in flat emission spectra; however, the adiabatic jets require a specific geometry. No re-acceleration process along the jet is needed for the electrons. We apply the models to observational data of the X-ray binary Cygnus X-1. Both models can be made consistent with the observations. The resulting ballistic jet is extremely narrow with a jet opening angle of only 5 arcsec. Its energy transport rate is small compared to the time-averaged jet power and therefore suggests the presence of non-radiating protons in the jet flow. The adiabatic jets require a strong departure from energy equipartition between the magnetic field and the relativistic electrons. These models also imply a jet power of two orders of magnitude higher than the Eddington limiting luminosity of a  10-M_⊙  black hole. The models put strong constraints on the physical conditions in the jet flows on scales well below achievable resolution limits.     

Galaxy shells and the structure of radio galaxies: Clues from Centaurus A (NGC 5128)

We show that the northern middle radio lobe of Cen A, an intriguing and much debated manifestation of radio lobe asymmetry, can be understood in terms of a direct interaction of the northern jet with a gaseous cloud associated with a stellar shell. This same basic mechanism was proposed earlier for the northern inner lobe, but new data allows a more detailed case to be made for the northern middle lobe. Although such an interaction can presently be demonstrated only for Cen A, the nearest radio galaxy, it is likely to be a fairly common occurrence and it provides an alternative to models invoking episodic nuclear activity, possibly accompanied with jet precession, for radio galaxies with multiple lobes and S-shapes. This proposed scenario may also play a key role in the origin of prominent radio galaxy morphological classes, such as the Wide-Angle-Tail sources and the Z-symmetric X-shaped radio sources. The strong tendency for radio lobes to be more distorted in double radio sources with jets that are in closer alignment with the optical major axis of the host elliptical galaxy can likewise be understood in terms of jet–shell interactions. In the frequent cases when jet activity is triggered by mergers of a large elliptical galaxy with a disk galaxy containing cold gas the impact of the gas associated with stellar shells upon the jets is likely to have significant manifestations.     

Active galactic nuclei jet mass loading and truncation by stellar winds

Active galactic nuclei can produce extremely powerful jets. While tightly collimated, the scale of these jets and the stellar density at galactic centres implies that there will be many jet/star interactions, which can mass load the jet through stellar winds. Previous work employed modest wind mass outflow rates, but this does not apply when mass loading is provided by a small number of high mass-loss stars. We construct a framework for jet mass loading by stellar winds for a broader spectrum of wind mass-loss rates than has previously been considered. Given the observed stellar mass distributions in galactic centres, we find that even highly efficient (0.1 Eddington luminosity) jets from supermassive black holes of masses M _BH≲ 10⁴ M_⊙ are rapidly mass loaded and quenched by stellar winds. For  10⁴ M_⊙ < M _BH < 10⁸ M_⊙  , the quenching length of highly efficient jets is independent of the jet's mechanical luminosity. Stellar wind mass loading is unable to quench efficient jets from more massive engines, but can account for the observed truncation of the inefficient M87 jet, and implies a baryon-dominated composition on scales ≳2 kpc therein even if the jet is initially pair plasma dominated.     

On the central core in MHD winds and jets

We analytically determine the structure of highly magnetized astrophysical jets at the origin in a region where the flow has been already collimated by an external medium, in both relativistic and non-relativistic regimes. We show that this can be achieved by solving a system of first-order ordinary differential equations that describe the transversal jet structure for a variety of external confining pressure profiles that collimate the jet to a near-cylindrical configuration. We obtain solutions for a central jet surrounded either by a self-similar wind or by an external pressure profile and derive the dependence of the velocity and the magnetic field strength along and across our jets. In particular, we find that the central core in a jet – the part of a flow with a nearly homogeneous magnetic field – must contain a poloidal field which is not much smaller than the critical value B _min. This allows us to determine the magnetic flux in a core which is much smaller than the total magnetic flux. We show that for such a small core flux the solutions with a magnetic field in a core much smaller than B _min are non-physical. For astrophysical objects the value of the critical magnetic field is quite large: 1 G for active galactic nuclei, 10¹⁰ G for gamma-ray bursts and 10⁻¹ G for young stellar objects. In a relativistic case for the core field greater than or of the order of B _min we show analytically that the plasma Lorentz factor must grow linearly with the cylindrical radius. For non-relativistic highly magnetized jets we propose that an oblique shock exists near the base of the jet so that the finite gas pressure plays an important role in force balance.     

The large-scale structure of FR-II radio sources

The large-scale flow produced by classical and relativistic jets in a uniform external medium is explored using a combination of general arguments and numerical simulations. We find that in both cases, jets with finite initial opening angles are recollimated by the high pressure in the cocoon and that the outer flow becomes approximately self-similar at large times. However, if the opening angle is significantly less than 20°, then there is an intermediate stage during which the working surface propagates at a constant speed, which is of the same order as that in the jet. The behaviour of the relativistic and classical jets is very similar, except that the relativistic jets generate lighter cocoons. Application of the model to Cygnus A gives estimates of the source age and advance speed which agree very well with spectral ageing observations. Quantitative estimates and general arguments suggest that the regularly spaced knots in the Cygnus A jet can be interpreted as shocks associated with reconfinement of an initially free jet, knot 3 of the Cygnus A jet being identified with the reflection point of the reconfinement shock. However, the model predicts too large an initial opening angle for the Cygnus A jets. It is possible that this discrepancy is due to our imposition of axisymmetry which allows the numerical jets to become much better collimated after the reconfinement than they would be in the three-dimensional case. Further study is needed to test this idea.     

Stability of magnetic equilibria in radio bubbles

Current-carrying flows, in the laboratory and in astrophysical jets, can form remarkably stable magnetic structures. Decades of experience show that such flows often build equilibria that reverse field directions, evolving to a magnetohydrodynamic (MHD) Taylor state, which has remarkable stability properties. We model jets and the magnetic bubbles they build as reversed-field pinch equilibria by assuming the driver current to be stiff in the MHD sense. Taking the jet current as rigid and a fixed function of position, we prove a theorem: that the same, simple MHD stability conditions guarantee stability, even after the jet turns off. This means that magnetic structures harbouring a massive inventory of magnetic energy can persist long after the building jet current has died away. These may be the relic radio 'fossils', 'ghost bubbles' or 'magnetic balloons' found in clusters. These equilibria, which are under magnetic tension, will evolve, retaining the stability properties from that state. The remaining fossil is not a disordered ball of magnetic fields, but a stable structure under tension, able to respond to the slings and arrows of outside forces. Typically their Alfvén speeds greatly exceed the cluster sound speed, and so they can keep out hot cluster plasma, leading to X-ray ghosts. Passing shocks cannot easily destroy them, but can energize and light them up anew at radio frequencies. Bubbles can rise in the hot cluster plasma, perhaps detaching from the parent radio galaxy but stable against Rayleigh–Taylor and other modes.     

Radio jet interactions with massive clouds

Three-dimensional simulations of light hydrodynamic jets are computed using the Zeus-3D code. We employ parameters corresponding to moderate to high power radio jets emerging through a galactic atmosphere or halo, and eventually crossing a tilted pressure matched interface with a hotter intracluster medium. These simulations aim the jets so that they hit massive dense clouds within the galactic halo. Such clouds are set up with radii several times that of the jet, and nominally correspond to giant molecular cloud complexes or small cannibalized galaxies. We find that powerful jets eventually disperse the clouds, but that, for the off-center collisions considered, non-axisymmetric instabilities are induced in those jets. Those instabilities grow faster for lower Mach number jets, and can produce disruptions substantially sooner than occurred in our earlier work on jets in the absence of collisions with massive clouds. Such interactions could be related to some Compact Steep Spectrum source morphologies. Very weak jets can be effectively halted by reasonably massive clouds, and this may have relevance for the paucity of radio jets in spiral galaxies. Slow, dense jets may be bent, yet remain stable for fairly extended times, thereby explaining some Wide-Angle-Tail and most “dog-leg” morphologies.     

The Dynamics of Stellar Jets

This paper discusses the formation of two-shock working surfaces in Herbig-Haro (HH) jets. These working surfaces can be formed either at the leading edge of the jet flow, or inside the body of the jet beam (as a result of variabilities in the jet flow velocity), and depending on the parameters of the flow can be either massless or mass conserving. It is shown that observations might indicate that these two regimes actually occur in some HH jets. This revised version was published online in July 2006 with corrections to the Cover Date.     

The effects of vigorous mixing in a convective model of zonal flow on the ice giants

Previous studies have used models of three-dimensional (3D) Boussinesq convection in a rotating spherical shell to explain the zonal flows on the gas giants, Jupiter and Saturn. In this paper we demonstrate that this approach can also generate flow patterns similar to those observed on the ice giants, Uranus and Neptune. The equatorial jets of Uranus and Neptune are often assumed to result from baroclinic cloud layer processes and have been simulated with shallow layer models. Here we show that vigorous, 3D convection in a spherical shell can produce the retrograde (westward) equatorial flows that occur on the ice giants as well as the prograde (eastward) equatorial flows of the gas giants. In our models, the direction of the equatorial jet depends on the ratio of buoyancy to Coriolis forces in the system. In cases where Coriolis forces dominate buoyancy, cylindrical Reynolds stresses drive prograde equatorial jets. However, as buoyancy forces approach and exceed Coriolis forces, the cylindrical nature of the flow is lost and 3D mixing homogenizes the fluid's angular momentum; the equatorial jet reverses direction, while strong prograde jets form in the polar regions. Although the results suggest that conditions involving strong atmospheric mixing are responsible for generating the zonal flows on the ice giants, our present models require roughly 100 and 10 times the internal heat fluxes observed on Uranus and Neptune, respectively.     

Three-Dimensional Simulations of Jets from Keplerian Disks: Stability Issues

Three-dimensional simulations of the time-dependent evolution of non-relativistic outflows from the surface of Keplerian accretion disks are presented. We investigate the outflow that arises from a magnetized accretion disk that is initially in hydrostatic balance with its surrounding cold corona. Our simulations show that jets maintain their long-term stability through a self-limiting process wherein the average Alfvénic Mach number within the jet is maintained to order unity. This is accomplished in at least two ways. First, poloidal magnetic field is concentrated along the central axis of the jet forming a `backbone' in which the Alfvén speed is sufficiently high to reduce the average jet Alfvénic Mach number to unity. Second, the onset of higher order Kelvin-Helmholtz `flute' modes (m ≥ 2) reduce the efficiency with which the jet material is accelerated, and transfer kinetic energy of the out flow into the stretched, poloidal field lines of the distorted jet. This too has the effect of increasing the Alfvén speed and thereby reducing the Alfvénic Mach number. The jet is able to survive the onset of the more destructive m=1 mode in this way.     

Models for jet power in elliptical galaxies: a case for rapidly spinning black holes

The power of jets from black holes is expected to depend on both the spin of the black hole and the structure of the accretion disc in the region of the last stable orbit. We investigate these dependencies using two different physical models for the jet power: the classical Blandford–Znajek (BZ) model and a hybrid model developed by Meier. In the BZ case, the jets are powered by magnetic fields directly threading the spinning black hole while in the hybrid model, the jet energy is extracted from both the accretion disc as well as the black hole via magnetic fields anchored to the accretion flow inside and outside the hole's ergosphere. The hybrid model takes advantage of the strengths of both the Blandford–Payne and BZ mechanisms, while avoiding the more controversial features of the latter. We develop these models more fully to account for general relativistic effects and to focus on advection-dominated accretion flows (ADAFs) for which the jet power is expected to be a significant fraction of the accreted rest mass energy.
We apply the models to elliptical galaxies, in order to see if these models can explain the observed correlation between the Bondi accretion rates and the total jet powers. For typical values of the disc viscosity parameter  α∼ 0.04 –0.3  and mass accretion rates consistent with ADAF model expectations, we find that the observed correlation requires   j ≳ 0.9  ; that is, it implies that the black holes are rapidly spinning. Our results suggest that the central black holes in the cores of clusters of galaxies must be rapidly rotating in order to drive jets powerful enough to heat the intracluster medium and quench cooling flows.     

A dynamo driven by zonal jets at the upper surface: Applications to giant planets

We present a dynamo mechanism arising from the presence of barotropically unstable zonal jet currents in a rotating spherical shell. The shear instability of the zonal flow develops in the form of a global Rossby mode, whose azimuthal wavenumber depends on the width of the zonal jets. We obtain self-sustained magnetic fields at magnetic Reynolds numbers greater than 10³. We show that the propagation of the Rossby waves is crucial for dynamo action. The amplitude of the axisymmetric poloidal magnetic field depends on the wavenumber of the Rossby mode, and hence on the width of the zonal jets. We discuss the plausibility of this dynamo mechanism for generating the magnetic field of the giant planets. Our results suggest a possible link between the topology of the magnetic field and the profile of the zonal winds observed at the surface of the giant planets. For narrow Jupiter-like jets, the poloidal magnetic field is dominated by an axial dipole whereas for wide Neptune-like jets, the axisymmetric poloidal field is weak.     

iShocks: X-ray binary jets with an internal shocks model

In the following paper, we present an internal shocks model, iShocks, for simulating a variety of relativistic jet scenarios; these scenarios can range from a single ejection event to an almost continuous jet, and are highly user configurable. Although the primary focus in the following paper is black hole X-ray binary jets, the model is scale and source independent and could be used for supermassive black holes in active galactic nuclei or other flows such as jets from neutron stars. Discrete packets of plasma (or 'shells') are used to simulate the jet volume. A two-shell collision gives rise to an internal shock, which acts as an electron re-energization mechanism. Using a pseudo-random distribution of the shell properties, the results show how for the first time it is possible to reproduce a flat/inverted spectrum (associated with compact radio jets) in a conical jet whilst taking the adiabatic energy losses into account. Previous models have shown that electron re-acceleration is essential in order to obtain a flat spectrum from an adiabatic conical jet: multiple internal shocks prove to be efficient in providing this re-energization. We also show how the high-frequency turnover/break in the spectrum is correlated with the jet power,  ν _b ∝ L ^∼0.6_W  , and the flat-spectrum synchrotron flux is correlated with the total jet power,   F _ν∝ L ^∼1.4_W  . Both the correlations are in agreement with previous analytical predictions.     

Deep jets on gas-giant planets

Three-dimensional numerical simulations of the atmospheric flow on giant planets using the primitive equations show that shallow thermal forcing confined to pressures near the cloud tops can produce deep zonal winds from the tropopause all the way down to the bottom of the atmosphere. These deep winds can attain speeds comparable to the zonal jet speeds within the shallow, forced layer; they are pumped by Coriolis acceleration acting on a deep meridional circulation driven by the shallow-layer eddies. In the forced layer, the flow reaches an approximate steady state where east-west eddy accelerations balance Coriolis accelerations acting on the meridional flow. Under Jupiter-like conditions, our simulations produce 25 to 30 zonal jets, similar to the number of jets observed on Jupiter and Saturn. The simulated jet widths correspond to the Rhines scale; this suggests that, despite the three-dimensional nature of the dynamics, the baroclinic eddies energize a quasi-two-dimensional inverse cascade modified by the β effect (where β is the gradient of the Coriolis parameter). In agreement with Jupiter, the jets can violate the barotropic and Charney-Stern stability criteria, achieving curvatures ^∂2u/∂y² of the zonal wind u with northward distance y up to 2β. The simulations exhibit a tendency toward neutral stability with respect to Arnol'd's second stability theorem in the upper troposphere, as has been suggested for Jupiter, although deviations from neutrality exist. When the temperature varies strongly with latitude near the equator, our simulations can also reproduce the stable equatorial superrotation with wind speeds greater than . Diagnostics show that barotropic eddies at low latitudes drive the equatorial superrotation. The simulations also broadly explain the distribution of jet-pumping eddies observed on Jupiter and Saturn. While idealized, these simulations therefore capture many aspects of the cloud-level flows on Jupiter and Saturn.