Heat-flux-driven Buoyancy Instability Research Articles

A new buoyancy instability in galaxy clusters due to streaming cosmic rays

ABSTRACT Active Galactic Nuclei (AGN) are believed to provide the energy that prevents runaway cooling of gas in the cores of galaxy clusters. However, how this energy is transported and thermalized throughout the Intracluster Medium (ICM) remains unclear. In recent work, we showed that streaming cosmic rays (CRs) destabilize sound waves in dilute ICM plasmas. Here, we show that CR streaming in the presence of gravity also destabilizes a pressure-balanced wave. We term this new instability the CR buoyancy instability (CRBI). In stark contrast to standard results without CRs, the pressure-balanced mode is highly compressible at short wavelengths due to CR streaming. Maximal growth rates are of order (pc/pg)β1/2ωff, where pc/pg is the ratio of CR pressure to thermal gas pressure, β is the ratio of thermal to magnetic pressure, and ωff is the free-fall frequency. The CRBI operates alongside buoyancy instabilities driven by background heat fluxes, i.e. the heat-flux-driven buoyancy instability (HBI) and the magneto-thermal instability (MTI). When the thermal mean free path lmfp is ≪ the gas scale height H, the HBI/MTI set the growth rate on large scales, while the CRBI sets the growth rate on small scales. Conversely, when lmfp ∼ H and (pc/pg)β1/2 ≳ 1, CRBI growth rates exceed HBI/MTI growth rates even on large scales. Our results suggest that CR-driven instabilities may be partially responsible for the sound waves/weak shocks and turbulence observed in galaxy clusters. CR-driven instabilities generated near radio bubbles may also play an important role redistributing AGN energy throughout clusters.

Grim: A Flexible, Conservative Scheme for Relativistic Fluid Theories

Abstract Hot, diffuse, relativistic plasmas such as sub-Eddington black-hole accretion flows are expected to be collisionless, yet are commonly modeled as a fluid using ideal general relativistic magnetohydrodynamics (GRMHD). Dissipative effects such as heat conduction and viscosity can be important in a collisionless plasma and will potentially alter the dynamics and radiative properties of the flow from that in ideal fluid models; we refer to models that include these processes as Extended GRMHD. Here we describe a new conservative code, grim, that enables all of the above and additional physics to be efficiently incorporated. grim combines time evolution and primitive variable inversion needed for conservative schemes into a single step using an algorithm that only requires the residuals of the governing equations as inputs. This algorithm enables the code to be physics agnostic as well as flexibility regarding time-stepping schemes. grim runs on CPUs, as well as on GPUs, using the same code. We formulate a performance model and use it to show that our implementation runs optimally on both architectures. grim correctly captures classical GRMHD test problems as well as a new suite of linear and nonlinear test problems with anisotropic conduction and viscosity in special and general relativity. As tests and example applications, we resolve the shock substructure due to the presence of dissipation, and report on relativistic versions of the magneto-thermal instability and heat flux driven buoyancy instability, which arise due to anisotropic heat conduction, and of the firehose instability, which occurs due to anisotropic pressure (i.e., viscosity). Finally, we show an example integration of an accretion flow around a Kerr black hole, using Extended GRMHD.

ON HELIUM MIXING IN QUASI-GLOBAL SIMULATIONS OF THE INTRACLUSTER MEDIUM

ABSTRACT The assumption of a spatially uniform helium distribution in the intracluster medium (ICM) can lead to biases in the estimates of key cluster parameters if composition gradients are present. The helium concentration profile in galaxy clusters is unfortunately not directly observable. Current models addressing the putative sedimentation are one-dimensional and parametrize the presence of magnetic fields in a crude way, ignoring the weakly collisional, magnetized nature of the medium. When these effects are considered, a wide variety of instabilities can play an important role in the plasma dynamics. In a series of recent papers, we have developed the local, linear theory of these instabilities and addressed their nonlinear development with a modified version of Athena. Here, we extend our study by developing a quasi-global approach that we use to simulate the mixing of helium as induced by generalizations of the heat-flux-driven buoyancy instability (HBI) and the magnetothermal instability, which feed off thermal and composition gradients. In the inner region of the ICM, mixing can occur over a few gigayears, after which the average magnetic field inclination angle is ∼30°–50°, resulting in an averaged Spitzer parameter higher by about 20% than the value obtained in homogeneous simulations. In the cluster outskirts the instabilities are rather inefficient, due to the shallow gradients. This suggests that composition gradients in cluster cores might be shallower than one-dimensional models predict. More quantitative statements demand more refined models that can incorporate the physics driving the sedimentation process and simultaneously account for the weakly collisional nature of the plasma.

PLASMA INSTABILITIES IN THE CONTEXT OF CURRENT HELIUM SEDIMENTATION MODELS: DYNAMICAL IMPLICATIONS FOR THE ICM IN GALAXY CLUSTERS

Understanding whether Helium can sediment to the core of galaxy clusters is important for a number of problems in cosmology and astrophysics. All current models addressing this question are one-dimensional and do not account for the fact that magnetic fields can effectively channel ions and electrons, leading to anisotropic transport of momentum, heat, and particle diffusion in the weakly collisional intracluster medium (ICM). This anisotropy can lead to a wide variety of instabilities, which could be relevant for understanding the dynamics of heterogeneous media. In this paper, we consider the radial temperature and composition profiles as obtained from a state-of-the-art Helium sedimentation model and analyze its stability properties. We find that the associated radial profiles are unstable, to different kinds of instabilities depending on the magnetic field orientation, at all radii. The fastest growing modes are usually related to generalizations of the Magnetothermal Instability (MTI) and the Heat-flux-driven Buoyancy Instability (HBI) which operate in heterogeneous media. We find that the effect of sedimentation is to increase (decrease) the predicted growth rates in the inner (outer) cluster region. The unstable modes grow fast compared to the sedimentation timescale. This suggests that the composition gradients as inferred from sedimentation models, which do not fully account for the anisotropic character of the weakly collisional environment, might not be very robust. Our results emphasize the subtleties involved in understanding the gas dynamics of the ICM and argue for the need of a comprehensive approach to address the issue of Helium sedimentation beyond current models.

ROLE OF MAGNETIC FIELD STRENGTH AND NUMERICAL RESOLUTION IN SIMULATIONS OF THE HEAT-FLUX-DRIVEN BUOYANCY INSTABILITY

The role played by magnetic fields in the intracluster medium (ICM) of galaxy clusters is complex. The weakly collisional nature of the ICM leads to thermal conduction that is channelled along field lines. This anisotropic heat conduction profoundly changes the stability of the ICM atmosphere, with convective stabilities being driven by temperature gradients of either sign. Here, we employ the Athena magnetohydrodynamic code to investigate the local non-linear behavior of the heat-flux driven buoyancy instability (HBI), relevant in the cores of cooling-core clusters where the temperature increases with radius. We study a grid of 2-d simulations that span a large range of initial magnetic field strengths and numerical resolutions. For very weak initial fields, we recover the previously known result that the HBI wraps the field in the horizontal direction thereby shutting off the heat flux. However, we find that simulations which begin with intermediate initial field strengths have a qualitatively different behavior, forming HBI-stable filaments that resist field-line wrapping and enable sustained vertical conductive heat flux at a level of 10--25% of the Spitzer value. While astrophysical conclusions regarding the role of conduction in cooling cores require detailed global models, our local study proves that systems dominated by HBI do not necessarily quench the conductive heat flux.

THE STABILITY OF WEAKLY COLLISIONAL PLASMAS WITH THERMAL AND COMPOSITION GRADIENTS

Over the last decade, substantial efforts have been devoted to understanding the stability properties, transport phenomena, and long-term evolution of weakly-collisional, magnetized plasmas which are stratified in temperature. These studies have improved our understanding of the physics governing the intra-cluster medium (ICM), but assumed that ICM is a homogeneous. This, however, might not be a good approximation if heavy elements sediment in the inner region of the galaxy cluster. In this paper, we analyze the stability of a weakly-collisional, magnetized plane-parallel atmosphere which is stratified in both temperature and composition. This allows us to discuss for the first time the dynamics of weakly-collisional environments where heat conduction, momentum transport, and ion-diffusion are anisotropic with respect to the direction of the magnetic field. We show that, depending on the relative signs and magnitudes of the gradients in the temperature and the mean molecular weight, the plasma can be subject to a wide variety of unstable modes which include modifications to the magnetothermal instability (MTI), the heat-flux-driven buoyancy instability (HBI), and overstable gravity modes previously studied in homogeneous media. We discuss the astrophysical implications of our findings for a representative galaxy cluster where helium has sedimented. Our findings suggest that the core insulation that results from the magnetic field configurations that arise as a natural consequence of the HBI, which would be MTI stable in a homogeneous medium, could be alleviated if the mean molecular weight gradient is steep enough, i.e., $(\nabla \mu)/\mu > (\nabla T)/T$. This study constitutes a first step toward understanding the interaction between magnetic turbulence and the diffusion of heavy elements in the ICM. (abridged)

The dynamics of Rayleigh-Taylor stable and unstable contact discontinuities with anisotropic thermal conduction

We study the effects of anisotropic thermal conduction along magnetic field lines on an accelerated contact discontinuity in a weakly collisional plasma. We first perform a linear stability analysis similar to that used to derive the Rayleigh-Taylor instability (RTI) dispersion relation. We find that anisotropic conduction is only important for compressible modes, as incompressible modes are isothermal. Modes grow faster in the presence of anisotropic conduction, but growth rates do not change by more than a factor of order unity. We next run fully non-linear numerical simulations of a contact discontinuity with anisotropic conduction. The non-linear evolution can be thought of as a superposition of three physical effects: temperature diffusion due to vertical conduction, the RTI, and the heat flux driven buoyancy instability (HBI). In simulations with RTI-stable contact discontinuities, the temperature discontinuity spreads due to vertical heat conduction. This occurs even for initially horizontal magnetic fields due to the initial vertical velocity perturbation and numerical mixing across the interface. The HBI slows this temperature diffusion by reorienting initially vertical magnetic field lines to a more horizontal geometry. In simulations with RTI-unstable contact discontinuities, the dynamics are initially governed by temperature diffusion, but the RTI becomes increasingly important at late times. We discuss the possible application of these results to supernova remnants, solar prominences, and cold fronts in galaxy clusters.

The HBI in a quasi-global model of the intracluster medium

In this paper, we investigate how convective instabilities influence heat conduction in the intracluster medium (ICM) of cool-core galaxy clusters. The ICM is a high-beta, weakly collisional plasma in which the transport of momentum and heat is aligned with the magnetic field. The anisotropy of heat conduction, in particular, gives rise to instabilities that can access energy stored in a temperature gradient of either sign. We focus on the heat-flux-driven buoyancy instability (HBI), which feeds on the outwardly increasing temperature profile of cluster cool cores. Our aim is to elucidate how the global structure of a cluster impacts on the growth and morphology of the linear HBI modes when in the presence of Braginskii viscosity, and ultimately on the ability of the HBI to thermally insulate cores. We employ an idealized quasi-global model, the plane-parallel atmosphere, which captures the essential physics – e.g. the global radial profile of the cluster – while letting the problem remain analytically tractable. Our main result is that the dominant HBI modes are localized to the innermost (≲20 per cent) regions of cool cores. It is then probable that, in the non-linear regime, appreciable field-line insulation will be similarly localized. Thus, while radio-mode feedback appears necessary in the central few tens of kpc, heat conduction may be capable of offsetting radiative losses throughout most of a cool core over a significant fraction of the Hubble time. Finally, our linear solutions provide a convenient numerical test for the non-linear codes that simulate the saturation of such convective instabilities in the presence of anisotropic transport.

The effects of anisotropic viscosity on turbulence and heat transport in the intracluster medium

In the intracluster medium (ICM) of galaxy clusters, heat and momentum are transported almost entirely along (but not across) magnetic field lines. We perform the first fully self-consistent Braginskii magnetohydrodynamics (MHD) simulations of galaxy clusters including both of these effects. Specifically, we perform local and global simulations of the magnetothermal instability (MTI) and the heat-flux-driven buoyancy instability (HBI) and assess the effects of viscosity on their saturation and astrophysical implications. We find that viscosity has only a modest effect on the saturation of the MTI. As in previous calculations, we find that the MTI can generate nearly sonic turbulent velocities in the outer parts of galaxy clusters, although viscosity somewhat suppresses the magnetic field amplification. At smaller radii in cool-core clusters, viscosity can decrease the linear growth rates of the HBI. However, it has less of an effect on the HBI’s non-linear saturation, in part because three-dimensional interchange motions (magnetic flux tubes slipping past each other) are not damped by anisotropic viscosity. In global simulations of cool-core clusters, we show that the HBI robustly inhibits radial thermal conduction and thus precipitates a cooling catastrophe. The effects of viscosity are, however, more important for higher entropy clusters. We argue that viscosity can contribute to the global transition of cluster cores from cool-core to non-cool-core states: additional sources of intracluster turbulence, such as can be produced by active galactic nuclei feedback or galactic wakes, suppress the HBI, heating the cluster core by thermal conduction; this makes the ICM more viscous, which slows the growth of the HBI, allowing further conductive heating of the cluster core and a transition to a non-cool-core state.

Dynamical stability of a thermally stratified intracluster medium with anisotropic momentum and heat transport

In weakly-collisional plasmas such as the intracluster medium (ICM), heat and momentum transport become anisotropic with respect to the local magnetic field direction. Anisotropic heat conduction causes the slow magnetosonic wave to become buoyantly unstable to the magnetothermal instability (MTI) when the temperature increases in the direction of gravity and to the heat-flux--driven buoyancy instability (HBI) when the temperature decreases in the direction of gravity. The local changes in magnetic field strength that attend these instabilities cause pressure anisotropies that viscously damp motions parallel to the magnetic field. In this paper we employ a linear stability analysis to elucidate the effects of anisotropic viscosity (i.e. Braginskii pressure anisotropy) on the MTI and HBI. By stifling the convergence/divergence of magnetic field lines, pressure anisotropy significantly affects how the ICM interacts with the temperature gradient. Instabilities which depend upon the convergence/divergence of magnetic field lines to generate unstable buoyant motions (the HBI) are suppressed over much of the wavenumber space, whereas those which are otherwise impeded by field-line convergence/divergence (the MTI) are strengthened. As a result, the wavenumbers at which the HBI survives largely unsuppressed in the ICM have parallel components too small to rigorously be considered local. This is particularly true as the magnetic field becomes more and more orthogonal to the temperature gradient. In contrast, the fastest-growing MTI modes are unaffected by anisotropic viscosity. However, we find that anisotropic viscosity couples slow and Alfven waves in such a way as to buoyantly destabilise Alfvenic fluctuations when the temperature increases in the direction of gravity. Consequently, many wavenumbers previously considered MTI-stable or slow-growing are in fact maximally unstable. (abridged)

Anisotropic dissipative effects on the buoyancy instability with background heat flux

The linear buoyancy instability in magnetized plasmas is investigated in the presence of anisotropic resistivity and viscosity by taking into account the background heat flux. The magnetic field is assumed to be homogeneous and has both horizontal and vertical components. The heat is primarily transported along the magnetic force lines when the gyro radius is much less than the mean collision free path. The Hall term is examined first and shows a damping effect on the magnetothermal instability. The heat-flux-driven buoyancy instability (HBI) is then investigated by taking into account the parallel resistivity (PR), cross-field resistivity (CR), and the anisotropic viscosity. The general dispersion relation (DR) is derived and discussed in several special cases. We show that only the CR and viscosity exert effects on the DR in the first case. The critical condition for the occurrence of HBI is modified by the CR coupled with the viscosity and the value of the instability growth rate is diminished by them. The effects due to the PR (resp. viscosity) on the HBI are examined next. The PR (resp. viscosity) is shown to alter not only the growth rate but also the instability criterion. There exists an unstable mode when the temperature decreases in the direction of gravity while this case is proven to be magnetothermally stable in the ideal magnetohydrodynamic limit. A new unstable mode is solely induced by the presence of PR (resp. viscosity). When the PR and CR are both taken into account, the resistivity is shown to induce a damping mode rather than an instability. Finally, considering the PR and viscosity simultaneously, it is found that a new unstable mode is excited when the PR is not equal to the viscosity, or else, dissipation effects do not alter the instability criterion and just cut down the growth rate.

Can conduction induce convection? On the non-linear saturation of buoyancy instabilities in dilute plasmas

We study the effects of anisotropic thermal conduction on low-collisionality, astrophysical plasmas using two and three-dimensional magnetohydrodynamic simulations. For weak magnetic fields, dilute plasmas are buoyantly unstable for either sign of the temperature gradient: the heat-flux-driven buoyancy instability (HBI) operates when the temperature increases with radius while the magnetothermal instability (MTI) operates in the opposite limit. In contrast to previous results, we show that, in the presence of a sustained temperature gradient, the MTI drives strong turbulence and operates as an efficient magnetic dynamo (akin to standard, adiabatic convection). Together, the turbulent and magnetic energies contribute up to ~10% of the pressure support in the plasma. In addition, the MTI drives a large convective heat flux, ~1.5% of rho c_s^3. These findings are robust even in the presence of an external source of strong turbulence. Our results on the nonlinear saturation of the HBI are consistent with previous studies but we explain physically why the HBI saturates quiescently by re-orienting the magnetic field (suppressing the conductive heat flux through the plasma), while the MTI saturates by generating sustained turbulence. We also systematically study how an external source of turbulence affects the saturation of the HBI: such turbulence can disrupt the HBI only on scales where the shearing rate of the turbulence is faster than the growth rate of the HBI. In particular, our results provide a simple mapping between the level of turbulence in a plasma and the effective isotropic thermal conductivity. We discuss the astrophysical implications of these findings, with a particular focus on the intracluster medium of galaxy clusters.

RADIATIVE AND DYNAMIC STABILITY OF A DILUTE PLASMA

We analyze the linear stability of a dilute, hot plasma, taking into account the effects of stratification and anisotropic thermal conduction. The work is motivated by attempts to understand the dynamics of the intracluster medium in galaxy clusters. We show that magnetic field configurations that nominally stabilize either the heat-flux driven buoyancy instability (associated with a positive thermal gradient) or the magnetothermal instability (negative thermal gradient) can lead to previously unrecognized g-mode overstabilities. The driving source of the overstability is either radiative cooling (positive temperature gradient) or the heat flux itself (negative temperature gradient). While the implications of these overstabilities have yet to be explored, we speculate that the cold fronts observed in many relaxed galaxy clusters may be related to their nonlinear evolution.

TURBULENCE IN GALAXY CLUSTER CORES: A KEY TO CLUSTER BIMODALITY?

We study the effects of externally imposed turbulence on the thermal properties of galaxy cluster cores, using three-dimensional numerical simulations including magnetic fields, anisotropic thermal conduction, and radiative cooling. The imposed "stirring" crudely approximates the effects of galactic wakes, waves generated by galaxies moving through the intracluster medium (ICM), and/or turbulence produced by a central active galactic nucleus. The simulated clusters exhibit a strong bimodality. Modest levels of turbulence, ~100 km/s (~10% of the sound speed), suppress the heat-flux-driven buoyancy instability (HBI), resulting in an isotropically tangled magnetic field and a quasi-stable, high entropy, thermal equilibrium with no cooling catastrophe. Thermal conduction dominates the heating of the cluster core, but turbulent mixing is critical because it suppresses the HBI and (to a lesser extent) the thermal instability. Lower levels of turbulent mixing (approximately less than 100 km/s) are insufficient to suppress the HBI, rapidly leading to a thermal runaway and a cool-core cluster. Remarkably, then, small fluctuations in the level of turbulence in galaxy cluster cores can initiate transitions between cool-core (low entropy) and non cool-core (high entropy) states.

Could AGN outbursts transform cool core clusters?

The origin of the bimodality in cluster core entropy is still unknown. At the same time, recent work has shown that thermal conduction in clusters is likely a time-variable phenomenon. We consider if time-variable conduction and AGN outbursts could be responsible for the cool-core (CC), non cool-core (NCC) dichotomy. We show that strong AGN heating can bring a CC cluster to a NCC state, which can be stably maintained by conductive heating from the cluster outskirts. On the other hand, if conduction is shut off by the heat-flux driven buoyancy instability, then the cluster will cool to the CC state again, where it is stabilized by low-level AGN heating. Thus, the cluster cycles between CC and NCC states. In contrast with massive clusters, we predict the CC/NCC bimodality should vanish in groups, due to the lesser role of conductive heating there. We find tentative support from the distribution of central entropy in groups, though firm conclusions require a larger sample carefully controlled for selection effects.

ANISOTROPIC THERMAL CONDUCTION AND THE COOLING FLOW PROBLEM IN GALAXY CLUSTERS

We examine the long-standing cooling flow problem in galaxy clusters with 3D MHD simulations of isolated clusters including radiative cooling and anisotropic thermal conduction along magnetic field lines. The central regions of the intracluster medium (ICM) can have cooling timescales of ~200 Myr or shorter--in order to prevent a cooling catastrophe the ICM must be heated by some mechanism such as AGN feedback or thermal conduction from the thermal reservoir at large radii. The cores of galaxy clusters are linearly unstable to the heat-flux-driven buoyancy instability (HBI), which significantly changes the thermodynamics of the cluster core. The HBI is a convective, buoyancy-driven instability that rearranges the magnetic field to be preferentially perpendicular to the temperature gradient. For a wide range of parameters, our simulations demonstrate that in the presence of the HBI, the effective radial thermal conductivity is reduced to less than 10% of the full Spitzer conductivity. With this suppression of conductive heating, the cooling catastrophe occurs on a timescale comparable to the central cooling time of the cluster. Thermal conduction alone is thus unlikely to stabilize clusters with low central entropies and short central cooling timescales. High central entropy clusters have sufficiently long cooling times that conduction can help stave off the cooling catastrophe for cosmologically interesting timescales.

BUOYANCY INSTABILITIES IN GALAXY CLUSTERS: CONVECTION DUE TO ADIABATIC COSMIC RAYS AND ANISOTROPIC THERMAL CONDUCTION

Using a linear stability analysis and two and three-dimensional nonlinear simulations, we study the physics of buoyancy instabilities in a combined thermal and relativistic (cosmic ray) plasma, motivated by the application to clusters of galaxies. We argue that cosmic ray diffusion is likely to be slow compared to the buoyancy time on large length scales, so that cosmic rays are effectively adiabatic. If the cosmic ray pressure $p_{cr}$ is $\gtrsim 25 %$ of the thermal pressure, and the cosmic ray entropy ($p_{\rm cr}/\rho^{4/3}$; $\rho$ is the thermal plasma density) decreases outwards, cosmic rays drive an adiabatic convective instability analogous to Schwarzschild convection in stars. Global simulations of galaxy cluster cores show that this instability saturates by reducing the cosmic ray entropy gradient and driving efficient convection and turbulent mixing. At larger radii in cluster cores, the thermal plasma is unstable to the heat flux-driven buoyancy instability (HBI), a convective instability generated by anisotropic thermal conduction and a background conductive heat flux. Cosmic-ray driven convection and the HBI may contribute to redistributing metals produced by Type 1a supernovae in clusters. Our calculations demonstrate that adiabatic simulations of galaxy clusters can artificially suppress the mixing of thermal and relativistic plasma; anisotropic thermal conduction allows more efficient mixing, which may contribute to cosmic rays being distributed throughout the cluster volume.

Nonlinear Simulations of the Heat-Flux-driven Buoyancy Instability and Its Implications for Galaxy Clusters

In low collisionality plasmas heat flows almost exclusively along magnetic field lines, and the condition for stability to convection is modified from the standard Schwarzschild criterion. We present local two and three-dimensional simulations of a new heat flux driven buoyancy instability (the HBI) that occurs when the temperature in a plasma decreases in the direction of gravity. We find that the HBI drives a convective dynamo that amplifies an initially weak magnetic field by a factor of ~20. In simulations that begin with the magnetic field aligned with the temperature gradient, the HBI saturates by rearranging the magnetic field lines to be almost purely perpendicular to the initial temperature gradient. This magnetic field reorientation results in a net heat flux through the plasma that is less than 1% of the field-free (Spitzer) value. We show that the HBI is likely to be present in the cool cores of clusters of galaxies between ~0.1-100 kpc, where the temperature increases outwards. The saturated state of the HBI suggests that inward thermal conduction from large radii in clusters is unlikely to solve the cooling flow problem. Finally, we also suggest that the HBI may contribute to suppressing conduction across cold fronts in galaxy clusters.