Magnetic Buoyancy Instability Research Articles

Non-linear magnetic buoyancy instability and turbulent dynamo

ABSTRACT Stratified discs with strong horizontal magnetic fields, are susceptible to magnetic buoyancy instability (MBI). Modifying the magnetic field and gas distributions, this can play an important role in galactic evolution. The MBI and the Parker instability, in which MBI is exacerbated by cosmic rays, are often studied using an imposed magnetic field. However, in galaxies and accretion discs, the magnetic field is continuously replenished by a large-scale dynamo action. Using non-ideal MHD equations, we model a section of the galactic disc (we neglect rotation and cosmic rays considered elsewhere), in which the large-scale field is generated by an imposed α-effect of variable intensity to explore the interplay between dynamo instability and MBI. The system evolves through three distinct phases: the linear (kinematic) dynamo stage, the onset of linear MBI when the magnetic field becomes sufficiently strong and the non-linear, statistically steady state. Non-linear effects associated with the MBI introduce oscillations which do not occur when the field is produced by the dynamo alone. The MBI initially accelerates the magnetic field amplification but the growth is quenched by the vertical motions produced by MBI. We construct a 1D model, which replicates all significant features of 3D simulations to confirm that magnetic buoyancy alone can quench the dynamo and is responsible for the magnetic field oscillations. Unlike similar results obtained with an imposed magnetic field, the non-linear interactions do not reduce the gas scale height, so the consequences of the magnetic buoyancy depend on how the magnetic field is maintained.

Joint Action and Competition between Centrifugal, Magnetorotational, and Magnetic Buoyancy Instabilities

Instabilities driven by some combination of rotation, velocity shear, and magnetic field in a stratified fluid under gravity play an important role in many astrophysical settings. Of particular note are the centrifugal instability, the magnetorotational instability, and magnetic buoyancy instability. Here, we consider a Cartesian model of an equatorial region incorporating all the physical ingredients necessary to study their competition. We investigate the linear instability to interchange (“axisymmetric”) modes of an inviscid, perfectly conducting, isothermal gas, including the effects of rotation, velocity shear, and poloidal and toroidal magnetic fields. The stability problem can be reduced to a second-order boundary value problem, with the growth rate as the eigenvalue. We can make analytic progress through consideration of the physically relevant regime in which the transverse horizontal wavenumber k ≫ 1. Via a perturbation analysis, with 1/k as the small parameter, we can derive the growth rate and the spatial dependence of the eigenfunctions: the unstable modes are strongly localized in the vertical direction, being either wall modes (localized near a boundary of the domain) or body modes (localized in the interior). We describe the conditions under which the joint action of the separate instability mechanisms leads to enhancement or suppression of the instability. Our analytical results are supplemented by numerical solutions of the stability problem. The most unstable mode found analytically is typically in excellent agreement with that found numerically through consideration of a wide range of wavenumbers. Finally, we discuss how our results relate to the solar tachocline.

Effects of Emerging Bipolar Magnetic Regions in Mean-field Dynamo Model of Solar Cycles 23 and 24

We model the physical parameters of Solar Cycles 23 and 24 using a nonlinear dynamical mean-field dynamo model that includes the formation and evolution of bipolar magnetic regions (BMRs). The Parker-type dynamo model consists of a complete MHD system in the mean-field formulation: the 3D magnetic induction equation, and 2D momentum and energy equations in the anelastic approximation. The initialization of BMRs is modeled in the framework of Parker’s magnetic buoyancy instability. It defines the depths of BMR injections, which are typically located at the edge of the global dynamo waves. The distribution with longitude and latitude and the size of the initial BMR perturbations are taken from the NOAA database of active regions. The modeling results are compared with various observed characteristics of the solar cycles. Only the BMR perturbations located in the upper half of the convection zone lead to magnetic active regions on the solar surface. While the BMRs initialized in the lower part of the convection zone do not emerge on the surface, they still affect the global dynamo process. Our results show that BMRs can play a substantial role in the dynamo processes and affect the strength of the solar cycles. However, the data driven model shows that the BMR’s effect alone cannot explain the weak Cycle 24. This weak cycle and the prolonged preceding minimum of magnetic activity were probably caused by a decrease of the turbulent helicity in the bulk of the convection zone during the decaying phase of Cycle 23.

Shear-driven magnetic buoyancy in the solar tachocline: the mean electromotive force due to rotation

ABSTRACT The leading theoretical paradigm for the Sun’s magnetic cycle is an αω-dynamo process, in which a combination of differential rotation and turbulent, helical flows produces a large-scale magnetic field that reverses every 11 yr. Most αω solar dynamo models rely on differential rotation in the solar tachocline to generate a strong toroidal field. The most problematic part of such models is then the production of the large-scale poloidal field, via a process known as the α-effect. Whilst this is usually attributed to small-scale convective motions under the influence of rotation, the efficiency of this regenerative process has been called into question by some numerical simulations. Motivated by likely conditions within the tachocline, the aim of this paper is to investigate an alternative mechanism for the poloidal field regeneration, namely the magnetic buoyancy instability in a shear-generated, rotating magnetic layer. We use a local, fully compressible model in which an imposed vertical shear winds up an initially vertical magnetic field. The field ultimately becomes buoyantly unstable, and we measure the resulting mean electromotive force (EMF). For sufficiently rapid rotation, we find that a significant component of the mean EMF is aligned with the direction of the mean magnetic field, which is the characteristic feature of the classical αω-dynamo model. Our results therefore suggest that magnetic buoyancy could contribute directly to the generation of large-scale poloidal field in the Sun.

Numerical Simulation of Solar Magnetic Flux Emergence Using the AMR–CESE–MHD Code

Magnetic flux emergence from the solar interior to the atmosphere is believed to be a key process in the formation of solar active regions and driving solar eruptions. Due to the limited capabilities of observations, the flux emergence process is commonly studied using numerical simulations. In this paper, we develop a numerical model to simulate the emergence of a twisted magnetic flux tube from the convection zone to the corona, using the AMR–CESE–MHD code, which is based on the conservation-element solution-element method, with adaptive mesh refinement. The results of our simulation agree with those of many previous studies with similar initial conditions, but by using different numerical codes. In the early stage, the flux tube rises from the convection zone, being driven by magnetic buoyancy, until it reaches close to the photosphere. The emergence is decelerated there, and with the piling up of the magnetic flux, the magnetic buoyancy instability is triggered, which allows the magnetic field to partially enter into the atmosphere. Meanwhile, two gradually separated polarity concentration zones appear in the photospheric layer, transporting the magnetic field and energy into the atmosphere through their vortical and shearing motions. Correspondingly, the coronal magnetic field is also reshaped into a sigmoid configuration, containing a thin current layer, which resembles the typical pre-eruptive magnetic configuration of an active region. Such a numerical framework of magnetic flux emergence as established will be applied to future investigations of how solar eruptions are initiated in flux emergence active regions.

Magnetic buoyancy instability and the anelastic approximation: regime of validity and relationship with compressible and Boussinesq descriptions

Magnetic buoyancy instability, which is of astrophysical importance, results from the influence of magnetic pressure variations on the density of a fluid in a gravitational field. It is inherently a compressible phenomenon and is, as such, fully described by the equations of compressible magnetohydrodynamics (MHD). For analytical and computational reasons, it is often convenient to study compressible MHD within simpler, asymptotically consistent reduced systems; the two most widely used result from the Boussinesq and anelastic approximations. Within the standard Boussinesq approximation of MHD, leading to the equations of Boussinesq magnetoconvection, magnetic buoyancy is excluded. It can, however, be included by a rescaling of the basic-state variables and by making further assumptions about the perturbation length scales. Within the anelastic approximation, no special measures are taken to incorporate magnetic buoyancy. It is, however, a priori unclear as to whether this neglect is justified, particularly in the light of the Boussinesq results. Our aims here are thus twofold. The first is to formulate the relationship between descriptions of magnetic buoyancy in the compressible, anelastic and Boussinesq systems. In so doing, we show that, under both the anelastic and Boussinesq approximations, magnetic buoyancy can be included either through a combination of a weak field and strong gradient, or, conversely, a strong field and weak gradient. Each has its own asymptotically consistent reduction, with dedicated governing equations. Our second aim is to address, through a linear stability analysis, under which conditions the standard anelastic system provides a faithful representation of magnetic buoyancy instability. For completeness, we also formulate the energy principle of ideal MHD within the anelastic framework, and demonstrate the relation with its fully compressible counterpart.

On Magnetic Inhibition Theory in Non-resistive Magnetohydrodynamic Fluids

We investigate why the non-slip boundary condition for the velocity, imposed in the direction of impressed magnetic fields, can contribute to the magnetic inhibition effect based on the nonhomogeneous magnetic Rayleigh--Taylor (abbr. NMRT) problem in non-resistive magnetohydrodynamic (abbr. MHD) fluids. Exploiting an infinitesimal method in Lagrangian coordinates, the idea of (equivalent) magnetic tension, and the differential version of magnetic flux conservation, we give an explanation of physical mechanism for the magnetic inhibition phenomenon in a non-resistive MHD fluid. Moreover, we find that the magnetic energy in the non-resistive MHD fluid depends on the displacement of fluid particles, and thus can be regarded as elastic potential energy. Motivated by this observation, we further use the well-known minimum potential energy principle to explain the physical meaning of the stability/instability criteria in the NMRT problem. As a result of the analysis, we further extend the results on the NMRT problem to the stratified MHD fluid case. We point out that our magnetic inhibition theory can be used to explain the inhibition phenomenon of other instabilities, such as thermal instability, magnetic buoyancy instability, and so on, by impressed magnetic fields in non-resistive MHD fluids.

Magnetic Buoyancy and Magnetorotational Instabilities in Stellar Tachoclines for Solar- and Antisolar-type Differential Rotation

Abstract We present results from an analytical model for magnetic buoyancy and rotational instabilities in full spherical shell stellar tachoclines that include rotation, differential rotation of either solar or antisolar type, and toroidal field. We find that in all cases, for latitudes where the tachocline vertical rotation gradient is positive, toroidal fields can be stored against magnetic buoyancy up to a limit that is proportional to the square root of the local vertical rotation gradient. For solar magnitude differential rotation, this limit is about 9 kG. For fixed percentage differential rotation, storage capacity varies linearly with the rotation rate. Faster rotators with the same percentage differential rotation can store larger fields, and slower rotators can store smaller fields. At latitudes where the vertical rotation gradient is negative, vigorous magnetorotational instability for even weak (≪1 kG) toroidal fields prevents such storage. We infer from these results that for stars with solar-type latitudinal differential rotation (fast equator, slow poles), any starspots present should be found in low latitudes, similar to the Sun. For antisolar differential rotation, any spots present should be found in mid- and high latitudes, perhaps with a peak of occurrence near 55°. These results hopefully provide some guidance for making and interpreting observations of stellar activity and differential rotation on stars with convection zones and tachoclines.

The Effect of Weak Resistivity and Weak Thermal Diffusion on Short-wavelength Magnetic Buoyancy Instability

Abstract Magnetic buoyancy instability in weakly resistive and thermally conductive plasma is an important mechanism of magnetic field expulsion in astrophysical systems. It is often invoked, e.g., in the context of the solar interior. Here, we revisit a problem introduc`ed by Gilman: the short-wavelength linear stability of a plane layer of compressible isothermal and weakly diffusive fluid permeated by a horizontal magnetic field of strength decreasing with height. In this physical setting, we investigate the effect of weak resistivity and weak thermal conductivity on the short-wavelength perturbations, localized in the vertical direction, and show that the presence of diffusion allows to establish the wavelength of the most unstable mode, undetermined in an ideal fluid. When diffusive effects are neglected, the perturbations are amplified at a rate that monotonically increases as the wavelength tends to zero. We demonstrate that, when the resistivity and thermal conduction are introduced, the wavelength of the most unstable perturbation is established and its scaling law with the diffusion parameters depends on gradients of the mean magnetic field, temperature, and density. Three main dynamical regimes are identified, with the wavelength of the most unstable mode scaling as either or or , where d is the layer thickness and is the ratio of the characteristic thermal diffusion velocity scale to the free-fall velocity. Our analytic results are backed up by a series of numerical solutions. The two-dimensional interchange modes are shown to dominate over three-dimensional ones when the magnetic field and/or temperature gradients are strong enough.

Magnetic Buoyancy and Rotational Instabilities in the Tachocline

Abstract We present results from an analytical model for magnetic buoyancy and rotational instabilities in a full spherical shell tachocline that includes rotation, differential rotation close to that observed helioseismically, and toroidal field. Perturbation solutions are found for the limit of large latitudinal wave number, a limit commonly used to maximize instability due to magnetic buoyancy. We find that at all middle and high latitudes vigorous rotational instability is induced by weak toroidal fields, particularly for high longitudinal wave number, even when the vertical rotation gradient is marginally stable without toroidal field. We infer that this instability will prevent much storage of toroidal fields in the tachocline at these latitudes, but could be responsible for the appearance of ephemeral active regions there. By contrast, the low-latitude vertical rotation gradient, opposite in sign to that at high latitudes, is not only stable itself but also prevents magnetic buoyancy instability until the peak toroidal field is raised above a threshold of about 9 kG at the equator, declining to zero where the vertical rotation gradient changes sign, at in our model. Thus this rotation gradient provides a previously unnoticed mechanism for storage of toroidal fields until they amplify by dynamo action to order 10 kG, whereupon they can overcome the rotation gradient to emerge as sunspots. These results provide a new explanation for why sunspots are seen only at low latitudes. The purely rotational instability at latitudes above 50°, even without toroidal fields, also suggests that the high-latitude tachocline should be much thicker, due to HD turbulence, than has been inferred for lower latitudes from helioseismic measurements.

Rayleigh-Taylor and Parker instabilities in MHD fluids

We present a brief survey of recent mathematical results on the stability and instability of magnetohydrodynamic flows in the presence of a gravitational field, which include the incompressible magnetic Rayleigh-Taylor instability, and the Parker instability (i.e., the magnetic buoyancy instability) in compressible magnetohydrodynamics. In particular, the stabilizing effect of some factors (e.g., steady magnetic fields, boundary conditions, domains) are analyzed. In addition, the mathematical results on the Rayleigh-Taylor instability problem in viscoelastic flows are discussed.

THEORETICAL LIMITS ON MAGNETIC FIELD STRENGTHS IN LOW-MASS STARS

ABSTRACT Observations have suggested that some low-mass stars have larger radii than predicted by 1D structure models. Some theoretical models have invoked very strong interior magnetic fields (of order 1 MG or more) as a possible cause of such large radii. Whether fields of that strength could in principle be generated by dynamo action in these objects is unclear, and we do not address the matter directly. Instead, we examine whether such fields could remain in the interior of a low-mass object for a significant amount of time, and whether they would have any other obvious signatures. First, we estimate the timescales for the loss of strong fields by magnetic buoyancy instabilities. We consider a range of field strengths and simple morphologies, including both idealized flux tubes and smooth layers of field. We confirm some of our analytical estimates using thin flux tube magnetohydrodynamic simulations of the rise of buoyant fields in a fully convective M-dwarf. Separately, we consider the Ohmic dissipation of such fields. We find that dissipation provides a complementary constraint to buoyancy: while small-scale, fibril fields might be regenerated faster than they rise, the dissipative heating associated with such fields would in some cases greatly exceed the luminosity of the star. We show how these constraints combine to yield limits on the internal field strength and morphology in low-mass stars. In particular, we find that for stars of 0.3 solar masses, no fields in flux tubes stronger than about 800 kG are simultaneously consistent with both constraints.

DIMMING AND CO ABSORPTION TOWARD THE AA TAU PROTOPLANETARY DISK: AN INFALLING FLOW CAUSED BY DISK INSTABILITY?

AA Tau, a classical T Tauri star in the Taurus cloud, has been the subject of intensive photometric monitoring for more than two decades due to its quasi-cyclic variation in optical brightness. Beginning in 2011, AA Tau showed another peculiar variation -- its median optical though near-IR flux dimmed significantly, a drop consistent with a 4-mag increase in visual extinction. It has stayed in the faint state since.Here we present 4.7um CO rovibrational spectra of AA Tau over eight epochs, covering an eleven-year time span, that reveal enhanced 12CO and 13CO absorption features in the $J_{\rm low}\leqslant$13 transitions after the dimming. These newly appeared absorptions require molecular gas along the line of sight with T~500 K and a column density of log (N12CO)~18.5 cm^{-2}, with line centers that show a constant 6 km s$^{-1}$ redshift. The properties of the molecular gas confirm an origin in the circumstellar material. We suggest that the dimming and absorption are caused by gas and dust lifted to large heights by a magnetic buoyancy instability. This material is now propagating inward, and on reaching the star within a few years will be observed as an accretion outburst.

MAGNETIC EFFECTS IN HOT JUPITER ATMOSPHERES

We present magnetohydrodynamic (MHD) simulations of the atmospheres of hot Jupiters ranging in temperature from 1100-1800K. Magnetic effects are negligible in atmospheres with temperatures $\lesssim$ 1400K. At higher temperatures winds are variable and in many cases, mean equatorial flows can become westward, opposite to their hydrodynamic counterparts. Ohmic dissipation peaks at temperatures $\sim$1500-1600K, depending on field strength, with maximum values $\sim 10^{18}$W at 10bar, substantially lower than previous estimates. Based on the limited parameter study done, this value can not be increased substantially with increasing winds, higher temperatures, higher field strengths, different boundary conditions or lower diffusivities. Although not resolved in these simulations there is modest evidence that a magnetic buoyancy instability may proceed in hot atmospheres.

Double-diffusive magnetic buoyancy instability in a quasi-two-dimensional Cartesian geometry

Magnetic buoyancy, believed to occur in the solar tachocline, is both an important part of large-scale solar dynamo models and the picture of how sunspots are formed. Given that in the tachocline region the ratio of magnetic diffusivity to thermal diffusivity is small it is important, for both the dynamo and sunspot formation pictures, to understand magnetic buoyancy in this regime. Furthermore, the tachocline is a region of strong shear and such investigations must involve structures that become buoyant in the double-diffusive regime which are generated entirely from a shear flow. In a previous study, we have illustrated that shear-generated doublediffusive magnetic buoyancy instability is possible in the tachocline. However, this study was severely limited due to the computational requirements of running three-dimensional magnetohydrodynamic simulations over diffusive time-scales. A more comprehensive investigation is required to fully understand the double-diffusive magnetic buoyancy instability and its dependency on a number of key parameters; such an investigation requires the consideration of a reduced model. Here we consider a quasi-two-dimensional model where all gradients in the x direction are set to zero. We show how the instability is sensitive to changes in the thermal diffusivity and also show how different initial configurations of the forced shear flow affect the behaviour of the instability. Finally, we conclude that if the tachocline is thinner than currently stated then the double-diffusive magnetic buoyancy instability can more easily occur.

SHORT-WAVELENGTH MAGNETIC BUOYANCY INSTABILITY

Magnetic buoyancy instability plays an important role in the evolution of astrophysical magnetic fields. Here we revisit the problem introduced by \citet{Gilman_1970} of the short wavelength linear stability of a plane layer of compressible isothermal fluid permeated by a horizontal magnetic field of strength decreasing with height. Dissipation of momentum and magnetic field is neglected. By the use of a Rayleigh-Schr\"odinger perturbation analysis, we explain in detail the limit in which the transverse horizontal wavenumber of the perturbation, denoted by $k$, is large (i.e.\ short horizontal wavelength) and show that the fastest growing perturbations become localized in the vertical direction as $k$ is increased. The growth rates are determined by a function of the vertical coordinate $z$ since, in the large $k$ limit, the eigenmodes are strongly localized in the vertical direction. We consider in detail the case of two-dimensional perturbations varying in the directions perpendicular to the magnetic field, which, for sufficiently strong field gradients, are the most unstable. The results of our analysis are backed up by comparison with a series of initial value problems. Finally we extend the analysis to three-dimensional perturbations.

Stably stratified magnetized stars in general relativity

We construct magnetized stars composed of a fluid stably stratified by entropy gradients in the framework of general relativity, assuming ideal magnetohydrodynamics and employing a barotropic equation of state. We first revisit basic equations for describing stably stratified stationary axisymmetric stars containing both poloidal and toroidal magnetic fields. As sample models, the magnetized stars considered by Ioka and Sasaki [23], inside which the magnetic fields are confined, are modified to the ones stably stratified. The magnetized stars newly constructed in this study are believed to be more stable than the existing relativistic models because they have both poloidal and toroidal magnetic fields with comparable strength, and magnetic buoyancy instabilities near the surface of the star, which can be stabilized by the stratification, are suppressed.

Magnetic buoyancy instabilities in the presence of magnetic flux pumping at the base of the solar convection zone

We perform idealised numerical simulations of magnetic buoyancy instabilities in a model of the solar tachocline. We introduce a simplified model of magnetic flux pumping in an upper layer (the convection zone), and study the effects of its inclusion on the evolution of buoyancy instabilities in a lower layer (the radiative interior). We study its effects on the instability of both a preconceived magnetic slab and of a shear-generated magnetic layer. In the former, we find that in the regime in which the downward pumping velocity is comparable with the Alfven speed of the magnetic layer, flux pumping is able to hold back the bulk of the magnetic field, with only small pockets of strong field able to rise into the upper layer. In simulations in which the magnetic layer is generated by shear, we find that the shear velocity is not necessarily required to exceed that of the pumping (therefore the kinetic energy of the shear is not required to exceed that of the overlying convection), for strong localised pockets of magnetic field to be produced which can rise into the upper layer. This is because magnetic flux pumping acts to store the field below the interface, allowing it to be amplified both by the shear, and by vortical fluid motions, until pockets of field can achieve sufficient strength to rise into the upper layer. In addition, we find that the interface between the two layers is a natural location for the production of strong vertical gradients in the magnetic field. If these gradients are sufficiently strong to allow the development of magnetic buoyancy instabilities, strong shear is not necessarily required to drive them (c.f. previous work by Vasil & Brummell). We find that the addition of magnetic flux pumping appears to be able to assist shear-driven magnetic buoyancy in producing strong flux concentrations that can rise into the convection zone from the radiative interior.

Modeling the Parker instability in a rotating plasma screw pinch

We analytically and numerically study the analogue of the Parker (magnetic buoyancy) instability in a uniformly rotating plasma screw pinch confined in a cylinder. Uniform plasma rotation is imposed to create a centrifugal acceleration, which mimics the gravity required for the classical Parker instability. The goal of this study is to determine how the Parker instability could be unambiguously identified in a weakly magnetized, rapidly rotating screw pinch, in which the rotation provides an effective gravity and a radially varying azimuthal field is controlled to give conditions for which the plasma is magnetically buoyant to inward motion. We show that an axial magnetic field is also required to circumvent conventional current driven magnetohydrodynamic (MHD) instabilities such as the sausage and kink modes that would obscure the Parker instability. These conditions can be realized in the Madison plasma Couette experiment (MPCX). Simulations are performed using the extended MHD code NIMROD for an isothermal compressible plasma model. Both linear and nonlinear regimes of the instability are studied, and the results obtained for the linear regime are compared with analytical results from a slab geometry. Based on this comparison, it is found that in a cylindrical pinch, the magnetic buoyancy mechanism dominates at relatively large Mach numbers (M > 5), while at low Mach numbers (M < 1), the instability is due to the curvature of magnetic field lines. At intermediate values of Mach number (1 < M < 5), the Coriolis force has a strong stabilizing effect on the plasma. A possible scenario for experimental demonstration of the Parker instability in MPCX is discussed.

The short-wavelength instability of magnetically buoyant layer

We revisit the problem introduced by Gilman (1970) and Acheson (1979) of linear stability of a plane layer of compressible fluid permeated by a horizontal magnetic field of magnitude decreasing with height with respect to short-wavelength two-dimensional perturbations varying in the directions perpendicular to the applied field. We show, that in the limit of large horizontal wave numbers the perturbations become strongly localised in the vertical direction. The motiavtion for this study is of astrophysical nature and comes from the common belief, that the magnetic buoyancy effects produce short-wavelength instabilities in the solar tachocline. We analyse the solar tachocline parameter regime to speculate about the strength of the magnetic field at the base of the solar convective zone and the time scales of the field variations induced by the magnetic buoyancy instability on the Sun.