Braginskii Viscosity Research Articles

Nonlinear Damping and Field-aligned Flows of Propagating Shear Alfvén Waves with Braginskii Viscosity

Braginskii magnetohydrodynamics (MHD) provides a more accurate description of many plasma environments than classical MHD since it actively treats the stress tensor using a closure derived from physical principles. Stress tensor effects nonetheless remain relatively unexplored for solar MHD phenomena, especially in nonlinear regimes. This paper analytically examines nonlinear damping and longitudinal flows of propagating shear Alfvén waves. Most previous studies of MHD waves in Braginskii MHD have considered the strict linear limit of vanishing wave perturbations. We show that those former linear results only apply to Alfvén wave amplitudes in the corona that are so small as to be of little interest, typically a wave energy less than 10−11 times the energy of the background magnetic field. For observed wave amplitudes, the Braginskii viscous dissipation of coronal Alfvén waves is nonlinear and a factor around 109 stronger than predicted by the linear theory. Furthermore, the dominant damping occurs through the parallel viscosity coefficient η 0, rather than the perpendicular viscosity coefficient η 2 in the linearized solution. This paper develops the nonlinear theory, showing that the wave energy density decays with an envelope . The damping length L d exhibits an optimal damping solution, beyond which greater viscosity leads to lower dissipation as the viscous forces self-organize the longitudinal flow to suppress damping. Although the nonlinear damping greatly exceeds the linear damping, it remains negligible for many coronal applications.

The role of physical and numerical viscosity in hydrodynamical instabilities

ABSTRACT The evolution of the Kelvin–Helmholtz Instability (KHI) is widely used to assess the performance of numerical methods. We employ this instability to test both the smoothed particle hydrodynamics (SPH) and the meshless finite mass (MFM) implementation in opengadget3. We quantify the accuracy of SPH and MFM in reproducing the linear growth of the KHI with different numerical and physical set-ups. Among them, we consider: (i) numerical induced viscosity, and (ii) physically motivated, Braginskii viscosity, and compare their effect on the growth of the KHI. We find that the changes of the inferred numerical viscosity when varying nuisance parameters such as the set-up or the number of neighbours in our SPH code are comparable to the differences obtained when using different hydrodynamical solvers, i.e. MFM. SPH reproduces the expected reduction of the growth rate in the presence of physical viscosity and recovers well the threshold level of physical viscosity needed to fully suppress the instability. In the case of galaxy clusters with a virial temperature of 3 × 107 K, this level corresponds to a suppression factor of ≈10−3 of the classical Braginskii value. The intrinsic, numerical viscosity of our SPH implementation in such an environment is inferred to be at least an order of magnitude smaller (i.e. ≈10−4), re-ensuring that modern SPH methods are suitable to study the effect of physical viscosity in galaxy clusters.

Effects of heat-flux vector and Braginskii viscosity on wave dissipation and instabilities in rotating gravitating anisotropic plasmas

Effects of heat-flux vector and Braginskii viscosity on wave dissipation and instabilities in rotating gravitating anisotropic plasmas

On the survival of cool clouds in the circumgalactic medium

ABSTRACT We explore the survival of cool clouds in multiphase circumgalactic media. We revisit the ‘cloud-crushing problem’ in a large survey of simulations including radiative cooling, self-shielding, self-gravity, magnetic fields, and anisotropic Braginskii conduction and viscosity (with saturation). We explore a wide range of parameters including cloud size, velocity, ambient temperature and density, and a variety of magnetic field configurations and cloud turbulence. We find that realistic magnetic fields and turbulence have weaker effects on cloud survival; the most important physics is radiative cooling and conduction. Self-gravity and self-shielding are important for clouds that are initially Jeans-unstable, but largely irrelevant otherwise. Non-self-gravitating, realistically magnetized clouds separate into four regimes: (1) at low column densities, clouds evaporate rapidly via conduction; (2) a ‘failed pressure confinement’ regime, where the ambient hot gas cools too rapidly to provide pressure confinement for the cloud; (3) an ‘infinitely long-lived’ regime, in which the cloud lifetime becomes longer than the cooling time of gas swept up in the leading bow shock, so the cloud begins to accrete and grow; and (4) a ‘classical cloud destruction’ regime, where clouds are eventually destroyed by instabilities. In the final regime, the cloud lifetime can exceed the naive cloud-crushing time owing to conduction-induced compression. However, small and/or slow-moving clouds can also evaporate more rapidly than the cloud-crushing time. We develop simple analytic models that explain the simulated cloud destruction times in this regime.

Braginskii viscosity on an unstructured, moving mesh accelerated with super-time-stepping

ABSTRACT We present a method for efficiently modelling Braginskii viscosity on an unstructured, moving mesh. Braginskii viscosity, i.e. anisotropic transport of momentum with respect to the direction of the magnetic field, is thought to be of prime importance for studies of the weakly collisional plasma that comprises the intracluster medium (ICM) of galaxy clusters. Here, anisotropic transport of heat and momentum has been shown to have profound consequences for the stability properties of the ICM. Our new method for modelling Braginskii viscosity has been implemented in the moving mesh code arepo. We present a number of examples that serve to test the implementation and illustrate the modified dynamics found when including Braginskii viscosity in simulations. These include (but are not limited to) damping of fast magnetosonic waves, interruption of linearly polarized Alfvén waves by the firehose instability, and the inhibition of the Kelvin–Helmholtz instability by Braginskii viscosity. An explicit update of Braginskii viscosity is associated with a severe time-step constraint that scales with (Δx)2, where Δx is the grid size. In our implementation, this restrictive time-step constraint is alleviated by employing second-order accurate Runge–Kutta–Legendre super-time-stepping. We envision including Braginskii viscosity in future large-scale simulations of Kelvin–Helmholtz unstable cold fronts in cluster mergers and AGN-generated bubbles in central cluster regions.

Effects of Anisotropic Viscosity on the Evolution of Active Galactic Nuclei Bubbles in Galaxy Clusters

Abstract The interaction between jets from active galactic nuclei (AGNs) and the intracluster medium (ICM) provides key constraints on the feeding and feedback of supermassive black holes. Much understanding about AGN feedback is gained from purely hydrodynamic models; however, whether such an approximation is adequate for the magnetized, weakly collisional ICM needs to be critically examined. For example, AGN-blown bubbles in hydrodynamic simulations are easily disrupted by fluid instabilities, making it difficult to explain the coherence of observed bubbles such as the northwest ghost bubble in Perseus. In order to investigate whether magnetic tension and viscosity in realistic conditions could preserve the bubble integrity, we performed the first Braginskii-magnetohydrodynamic simulations of jet-inflated bubbles in a medium with tangled magnetic field. We find that magnetic tension alone is insufficient to prevent bubble deformation due to large velocity shear at early stage of the evolution. Although unsuppressed anisotropic viscosity in tangled magnetic field can have similar effects as isotropic viscosity, when the pressure anisotropy is bounded by microscopic plasma instabilities, the level of viscosity is substantially limited, thereby failing to prevent bubble deformation as in the inviscid case. Our results suggest that Braginskii viscosity is unlikely to be the primary mechanism for suppressing the fluid instabilities for AGN bubbles, and it remains a challenging task to reproduce smooth and coherent bubbles as observed. Because the dynamical influence and heating of the ICM critically depend on the bubble morphology, our study highlights the fundamental role of “microphysics” on the macroscopic properties of AGN feedback processes.

Shearing-box simulations of MRI-driven turbulence in weakly collisional accretion discs.

We present a systematic shearing-box investigation of MRI-driven turbulence in a weakly collisional plasma by including the effects of an anisotropic pressure stress, i.e. anisotropic (Braginskii) viscosity. We constrain the pressure anisotropy (Δp) to lie within the stability bounds that would be otherwise imposed by kinetic microinstabilities. We explore a broad region of parameter space by considering different Reynolds numbers and magnetic-field configurations, including net vertical flux, net toroidal-vertical flux and zero net flux. Remarkably, we find that the level of turbulence and angular-momentum transport are not greatly affected by large anisotropic viscosities: the Maxwell and Reynolds stresses do not differ much from the MHD result. Angular-momentum transport in Braginskii MHD still depends strongly on isotropic dissipation, e.g., the isotropic magnetic Prandtl number, even when the anisotropic viscosity is orders of magnitude larger than the isotropic diffusivities. Braginskii viscosity nevertheless changes the flow structure, rearranging the turbulence to largely counter the parallel rate of strain from the background shear. We also show that the volume-averaged pressure anisotropy and anisotropic viscous transport decrease with increasing isotropic Reynolds number (Re); e.g., in simulations with net vertical field, the ratio of anisotropic to Maxwell stress (α A/α M) decreases from ~ 0.5 to ~ 0.1 as we move from Re ~ 103 to Re ~ 104, while 〈4πΔp/B 2〉 → 0. Anisotropic transport may thus become negligible at high Re. Anisotropic viscosity nevertheless becomes the dominant source of heating at large Re, accounting for ≳50% of the plasma heating. We conclude by briefly discussing the implications of our results for RIAFs onto black holes.

THE EFFECT OF ANISOTROPIC VISCOSITY ON COLD FRONTS IN GALAXY CLUSTERS

Cold fronts -- contact discontinuities in the intracluster medium (ICM) of galaxy clusters -- should be disrupted by Kelvin-Helmholtz (K-H) instabilities due to the associated shear velocity. However, many observed cold fronts appear stable. This opens the possibility to place constraints on microphysical mechanisms that stabilize them, such as the ICM viscosity and/or magnetic fields. We performed exploratory high-resolution simulations of cold fronts arising from subsonic gas sloshing in cluster cores using the grid-based Athena MHD code, comparing the effects of isotropic Spitzer and anisotropic Braginskii viscosity (expected in a magnetized plasma). Magnetized simulations with full Braginskii viscosity or isotropic Spitzer viscosity reduced by a factor f ~ 0.1 are both in qualitative agreement with observations in terms of suppressing K-H instabilities. The RMS velocity of turbulence within the sloshing region is only modestly reduced by Braginskii viscosity. We also performed unmagnetized simulations with and without viscosity and find that magnetic fields have a substantial effect on the appearance of the cold fronts, even if the initial field is weak and the viscosity is the same. This suggests that determining the dominant suppression mechanism of a given cold front from X-ray observations (e.g. viscosity or magnetic fields) by comparison with simulations is not straightforward. Finally, we performed simulations including anisotropic thermal conduction, and find that including Braginskii viscosity in these simulations does not significant affect the evolution of cold fronts; they are rapidly smeared out by thermal conduction, as in the inviscid case.

Viscous dissipation in 3D spine reconnection solutions

Aims. We consider the influence of viscous dissipation on “spine” reconnection solutions in coronal plasmas. It is known from 2D and 3D “fan” reconnection studies that viscous losses can be important. We extend these arguments to 3D “spine” reconnection solutions. Methods. Steady 3D spine reconnection models were constructed by time relaxation of the governing visco-resistive MHD equations. Scaling laws were derived that compare the relative importance of viscous and resistive damping. Results. It is shown that viscous dissipation in spine reconnection models can dominate resistive damping by many orders of magnitude. A similar conclusion, but with less severe implications, applies to current sheet “fan” solutions. These findings are not sensitive to whether classical or Braginskii viscosity is employed.

AGN feedback in clusters: Shock and sound heating

AbstractObservations support the view that feedback, in the form of radio outbursts from active nuclei in central galaxies, prevents catastro‐phic cooling of gas and rapid star formation in many groups and clusters of galaxies. Variations in jet power drive a succession of weak shocks that can heat regions close to the active galactic nuclei (AGN). On larger scales, shocks fade into sound waves. The Braginskii viscosity determines a well‐defined sound damping rate in the weakly magnetised intracluster medium (ICM) that can provide sufficient heating on larger scales. It is argued that weak shocks and sound dissipation are the main means by which radio AGN heat the ICM, in which case, the power spectrum of AGN outbursts plays a central role in AGN feedback. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

THEFERMIBUBBLES. II. THE POTENTIAL ROLES OF VISCOSITY AND COSMIC-RAY DIFFUSION IN JET MODELS

The origin of the Fermi bubbles recently detected by the Fermi Gamma-ray Space Telescope in the inner Galaxy is mysterious. In the companion paper Guo & Mathews (Paper I), we use hydrodynamic simulations to show that they could be produced by a recent powerful AGN jet event. Here we further explore this scenario to study the potential roles of shear viscosity and cosmic ray (CR) diffusion on the morphology and CR distribution of the bubbles. We show that even a relatively low level of viscosity (\mu_{visc} >~ 3 g cm^{-1} s^{-1}, or ~0.1% - 1% of Braginskii viscosity in this context) could effectively suppress the development of Kelvin-Helmholtz instabilities at the bubble surface, resulting in smooth bubble edges as observed. Furthermore, viscosity reduces circulating motions within the bubbles, which would otherwise mix the CR-carrying jet backflow near bubble edges with the bubble interior. Thus viscosity naturally produces an edge-favored CR distribution, an important ingredient to produce the observed flat gamma-ray surface brightness distribution. Generically, such a CR distribution often produces a limb-brightened gamma-ray intensity distribution. However, we show that by incorporating CR diffusion which is strongly suppressed across the bubble surface (as inferred from sharp bubble edges) but is close to canonical values in the bubble interior, we obtain a reasonably flat gamma-ray intensity profile. The similarity of the resulting CR bubble with the observed Fermi bubbles strengthens our previous result in Paper I that the Fermi bubbles were produced by a recent AGN jet event. Studies of the nearby Fermi bubbles may provide a unique opportunity to study the potential roles of plasma viscosity and CR diffusion on the evolution of AGN jets and bubbles.

Quasi-global galactic magnetorotational instability with Braginskii viscosity

We present a global-in-radius linear analysis of the axisymmetric magnetorotational instability (MRI) in a collisional magnetized plasma with Braginskii viscosity. For a galactic angular velocity profile $\Omega$ we obtain analytic solutions for three magnetic field orientations: purely azimuthal, purely vertical and slightly pitched (almost azimuthal). In the first two cases the Braginskii viscosity damps otherwise neutrally stable modes, and reduces the growth rate of the MRI respectively. In the final case the Braginskii viscosity makes the MRI up to $2\sqrt{2}$ times faster than its inviscid counterpart, even for \emph{asymptotically small} pitch angles. We investigate the transition between the Lorentz-force-dominated and the Braginskii viscosity-dominated regimes in terms of a parameter $\sim \Omega \nub/B^2$ where $\nub$ is the viscous coefficient and $B$ the Alfv\'en speed. In the limit where the parameter is small and large respectively we recover the inviscid MRI and the magnetoviscous instability (MVI). We obtain asymptotic expressions for the approach to these limits, and find the Braginskii viscosity can magnify the effects of azimuthal hoop tension (the growth rate becomes complex) by over an order of magnitude. We discuss the relevance of our results to the local approximation, galaxies and other magnetized astrophysical plasmas. Our results should prove useful for benchmarking codes in global geometries.

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.

Viscous effects in time-dependent planar reconnection

Context. Viscous dissipation is expected to play a significant part in energy release in solar flares, yet the role of viscosity in a weakly resistive plasma of the solar corona remains unclear. Aims. We attempt to clarify the role of viscous effects in magnetic reconnection by performing simulations of reconnection in planar periodic geometry in an incompressible viscous resistive plasma. Methods. We consider magnetic reconnection, driven by large-scale vortical flows. We use both the classical shear viscosity and the Braginskii form for the ion parallel viscosity in a magnetised plasma. We determine the scalings of the current sheet parameters and the global rates of resistive and viscous dissipation. We use steady-state exact solutions and scaling arguments to interpret the numerical results. Results. We show that, regardless of the form of viscosity, the resistive non-viscous analytical solutions for flux pile-up merging provide a very good approximation of the numerical results in the reconnecting current sheet. We find no evidence for a viscoresistive scale. Numerical results for a highly sheared magnetic field, however, appear to deviate from the analytical predictions in the case of the Braginskii viscosity.

ENERGY LOSSES BY ANISOTROPIC VISCOUS DISSIPATION IN TRANSIENT MAGNETIC RECONNECTION

Global energy losses associated with transient magnetic reconnection in a viscous resistive plasma are examined. The Braginskii stress tensor is used to model the plasma viscosity for conditions typical of the solar corona. Analytic arguments are used to show that the large-scale advective flows associated with magnetic merging are likely to generate significant viscous losses. It is pointed out that the development of a visco-resistive reconnection scale, predicted for the classical shear viscosity, is not expected in the more realistic case of the Braginskii viscosity. Numerical simulations of planar coalescence merging show that viscous losses should easily dominate resistive losses for physically plausible parameters in flaring regions. Our computations imply that flare-like rates exceeding 1029 erg s–1 can be achieved under plausible coronal conditions.

BUOYANT BUBBLES IN INTRACLUSTER GAS: EFFECTS OF MAGNETIC FIELDS AND ANISOTROPIC VISCOSITY

Recent observations by Chandra and XMM-Newton indicate there are complex structures at the cores of galaxy clusters, such as cavities and filaments. One plausible model for the formation of such structures is the interaction of radio jets with the intracluster medium (ICM). To investigate this idea, we use three-dimensional magnetohydrodynamic simulations including anisotropic (Braginskii) viscosity to study the effect of magnetic fields on the evolution and morphology of buoyant bubbles in the ICM. We investigate a range of different initial magnetic field geometries and strengths, and study the resulting x-ray surface brightness distribution for comparison to observed clusters. Magnetic tension forces and viscous transport along field lines tend to suppress instabilities parallel, but not perpendicular, to field lines. Thus, the evolution of the bubble depends strongly on the initial field geometry. We find toroidal field loops initially confined to the interior of the bubble are best able reproduce the observed cavity structures.

Enhanced phase mixing of Alfvén waves propagating in stratified and divergent coronal structures

Corrected analytical solutions describing the enhanced phase mixing of Alfven waves propagating in divergent stratified coronal structures are presented. These show that the enhanced phase mixing mechanism can dissipate Alfven waves at heights less than half that is predicted by the previous analytical solutions. The enhanced phase mixing of 0.1 Hz harmonic Alfven waves propagating in strongly divergent, H_b=5 Mm, stratified coronal structures, H_rho=50 Mm, can fulfill 100% of an active region heating requirement, by generating viscous heating fluxes of F_H~2100 J /m^2 /s. The Alfven waves in this configuration are fully dissipated within 20 Mm, which is six times lower than would occur as a result of standard phase mixing in uniform magnetic fields. This results in the heating scale height, s_H, being lowered by a factor of six, to less than half of an active regions density scale height. Using the corrected analytical solutions it was found that, for a given wave frequency, the generation of a heating scale height of s_H<=50 Mm, by enhanced phase mixing in strongly divergent magnetic fields, requires a shear viscosity eight orders of magnitude lower, than required by standard phase mixing in uniform magnetic fields. It was also found that the enhanced phase mixing of observable, 0.01 rads /s Alfven waves, in strongly divergent magnetic fields, H_b=5 Mm, can generate heating scale heights within a density scale height, H_rho=50 Mm, using classical Braginskii viscosity. It is therefore not necessary to invoke anomalous viscosity in corona, if phase mixing takes place in strongly divergent magnetic fields. This study shows that the importance of enhanced phase mixing as a mechanism for dissipating Alfven waves in the solar corona (a stratified and divergent medium), has been seriously underestimated.

Plasma Instabilities and Magnetic Field Growth in Clusters of Galaxies

We show that under very general conditions, cluster plasmas threaded by weak magnetic fields are subject to very fast growing plasma instabilities driven by the anisotropy of the plasma pressure (viscous stress) with respect to the local direction of the magnetic field. Such an anisotropy will naturally arise in any weakly magnetized plasma that has low collisionality and is subject to stirring. The magnetic field must be sufficiently weak for the instabilities to occur, viz., β Re1/2. The instabilities are captured by the extended MHD model with Braginskii viscosity. However, their growth rates are proportional to the wavenumber down to the ion gyroscale, so MHD equations with Braginskii viscosity are not well posed and a fully kinetic treatment is necessary. The instabilities can lead to magnetic fields in clusters being amplified from seed strength of ~10-18 G to dynamically important strengths of ~10 μG on cosmologically trivial timescales (~108 yr). The fields produced during the amplification stage are at scales much smaller than observed. Predicting the saturated field scale and structure will require a kinetic theory of magnetized cluster turbulence.

Radiative hydrodynamic modeling of the Bastille-Day flare (14 July, 2000)

A 1-D loop radiative hydrodynamic model that incorporates the effects of gravitational stratification, heat conduction, radiative losses, external heat input, presence of helium, and Braginskii viscosity is used to simulate elementary flare loops. The physical parameters for the input are taken from observations of the Bastille-Day flare of 2000 July 14. The present analysis shows that: (a) The obtained maximum values of the electron density can be considerably higher (4.2 X 10^{11} cm^{-3} or more) in the case of footpoint heating than in the case of apex heating (2.5 X 10^{11} cm^{-3}). (b) The average cooling time after the flare peak takes less time in the case of footpoint heating than in the case of apex heating. (c) The peak apex temperatures are significantly lower (by about 10 MK) for the case of footpoint heating than for apex heating (for the same average loop temperature of about 30 MK). This characteristic allows us to discriminate between different heating positioning. (d) In both cases (of apex and footpoint heating), the maximum obtained apex temperature T^{max} is practically independent of the heating duration sigma_{t}, but scales directly with the heating rate E_{H0}. (e) The maximum obtained densities at the loop apex, n_e^{max}, increase with the heating rate E_{H0} and heating duration sigma_{t} for both footpoint and apex heating. In Paper II we will use the outputs of these hydrodynamic simulations, which cover a wide range of the parameter space of heating rates and durations, as an input for forward-fitting of the multi-loop arcade of the Bastille-day flare. KEYWORDS: Sun: Flares -- Sun: Activity -- Sun: Corona

Magnetized Turbulent Dynamos in Protogalaxies

The prevailing theory for the origin of cosmic magnetic fields is that they have been amplified to their present values by the turbulent dynamo inductive action in the protogalactic and galactic medium. Up to now, in calculation of the turbulent dynamo, it has been customary to assume that there is no back reaction of the magnetic field on the turbulence, as long as the magnetic energy is less than the turbulent kinetic energy. This assumption leads to the kinematic dynamo theory. However, the applicability of this theory to protogalaxies is rather limited. The reason is that in protogalaxies the temperature is very high, and the viscosity is dominated by magnetized ions. As the magnetic field strength grows in time, the ion cyclotron time becomes shorter than the ion collision time, and the plasma becomes strongly magnetized. As a result, the ion viscosity becomes the Braginskii viscosity. Thus, in protogalaxies the back reaction sets in much earlier, at field strengths much lower than those which correspond to field-turbulence energy equipartition, and the turbulent dynamo becomes what we call the magnetized turbulent dynamo. In this paper we lay the theoretical groundwork for the magnetized turbulent dynamo. In particular, we predict that the magnetic energy growth rate in the magnetized dynamo theory is up to ten time larger than that in the kinematic dynamo theory. We also briefly discuss how the Braginskii viscosity can aid the development of the inverse cascade of magnetic energy after the energy equipartition is reached.