Hydromagnetic Turbulence Research Articles

A large-scale dynamo and magnetoturbulence in rapidly rotating core-collapse supernovae.

Magnetohydrodynamic turbulence is important in many high-energy astrophysical systems, where instabilities can amplify the local magnetic field over very short timescales. Specifically, the magnetorotational instability and dynamo action have been suggested as a mechanism for the growth of magnetar-strength magnetic fields (of 10(15) gauss and above) and for powering the explosion of a rotating massive star. Such stars are candidate progenitors of type Ic-bl hypernovae, which make up all supernovae that are connected to long γ-ray bursts. The magnetorotational instability has been studied with local high-resolution shearing-box simulations in three dimensions, and with global two-dimensional simulations, but it is not known whether turbulence driven by this instability can result in the creation of a large-scale, ordered and dynamically relevant field. Here we report results from global, three-dimensional, general-relativistic magnetohydrodynamic turbulence simulations. We show that hydromagnetic turbulence in rapidly rotating protoneutron stars produces an inverse cascade of energy. We find a large-scale, ordered toroidal field that is consistent with the formation of bipolar magnetorotationally driven outflows. Our results demonstrate that rapidly rotating massive stars are plausible progenitors for both type Ic-bl supernovae and long γ-ray bursts, and provide a viable mechanism for the formation of magnetars. Moreover, our findings suggest that rapidly rotating massive stars might lie behind potentially magnetar-powered superluminous supernovae.

Properties of p and f modes in hydromagnetic turbulence

With the ultimate aim of using the fundamental or $f$-mode to study helioseismic aspects of turbulence-generated magnetic flux concentrations, we use randomly forced hydromagnetic simulations of a piecewise isothermal layer in two dimensions with reflecting boundaries at top and bottom. We compute numerically diagnostic wavenumber-frequency diagrams of the vertical velocity at the interface between the denser gas below and the less dense gas above. For an Alfv\'en-to-sound speed ratio of about 0.1, a 5% frequency increase of the $f$-mode can be measured when $k_xH_{\rm p}=3$-$4$, where $k_x$ is the horizontal wavenumber and $H_{\rm p}$ is the pressure scale height at the surface. Since the solar radius is about 2000 times larger than $H_{\rm p}$, the corresponding spherical harmonic degree would be 6000-8000. For weaker fields, a $k_x$-dependent frequency decrease by the turbulent motions becomes dominant. For vertical magnetic fields, the frequency is enhanced for $k_xH_{\rm p}\approx4$, but decreased relative to its nonmagnetic value for $k_xH_{\rm p}\approx9$.

FANNING OUT OF THE SOLAR f -MODE IN THE PRESENCE OF NON-UNIFORM MAGNETIC FIELDS?

We show that in the presence of a harmonically varying magnetic field the fundamental or $f$-mode in a stratified layer is altered in such a way that it fans out in the diagnostic $k\omega$ diagram, but with mode power also within the fan. In our simulations, the surface is defined by a temperature and density jump in a piecewise isothermal layer. Unlike our previous work (Singh et al. 2014) where a uniform magnetic field was considered, we employ here a nonuniform magnetic field together with hydromagnetic turbulence at length scales much smaller than those of the magnetic fields. The expansion of the $f$-mode is stronger for fields confined to the layer below the surface. In some of those cases, the $k\omega$ diagram also reveals a new class of low frequency vertical stripes at multiples of twice the horizontal wavenumber of the background magnetic field. We argue that the study of the $f$-mode expansion might be a new and sensitive tool to determining subsurface magnetic fields with longitudinal periodicity.

QUENCHING AND ANISOTROPY OF HYDROMAGNETIC TURBULENT TRANSPORT

Hydromagnetic turbulence affects the evolution of large-scale magnetic fields through mean-field effects like turbulent diffusion and the α effect. For stronger fields, these effects are usually suppressed or quenched, and additional anisotropies are introduced. Using different variants of the test-field method, we determine the quenching of the turbulent transport coefficients for the forced Roberts flow, isotropically forced non-helical turbulence, and rotating thermal convection. We see significant quenching only when the mean magnetic field is larger than the equipartition value of the turbulence. Expressing the magnetic field in terms of the equipartition value of the quenched flows, we obtain for the quenching exponents of the turbulent magnetic diffusivity about 1.3, 1.1, and 1.3 for Roberts flow, forced turbulence, and convection, respectively. However, when the magnetic field is expressed in terms of the equipartition value of the unquenched flows, these quenching exponents become about 4, 1.5, and 2.3, respectively. For the α effect, the exponent is about 1.3 for the Roberts flow and 2 for convection in the first case, but 4 and 3, respectively, in the second. In convection, the quenching of turbulent pumping follows the same power law as turbulent diffusion, while for the coefficient describing the effect nearly the same quenching exponent is obtained as for α. For forced turbulence, turbulent diffusion proportional to the second derivative along the mean magnetic field is quenched much less, especially for larger values of the magnetic Reynolds number. However, we find that in corresponding axisymmetric mean-field dynamos with dominant toroidal field the quenched diffusion coefficients are the same for the poloidal and toroidal field constituents.

MAGNETIC PRANDTL NUMBER DEPENDENCE OF THE KINETIC-TO-MAGNETIC DISSIPATION RATIO

Using direct numerical simulations of three-dimensional hydromagnetic turbulence, either with helical or non-helical forcing, we show that the ratio of kinetic-to-magnetic energy dissipation always increases with the magnetic Prandtl number, i.e., the ratio of kinematic viscosity to magnetic diffusivity. This dependence can always be approximated by a power law, but the exponent is not the same in all cases. For non-helical turbulence, the exponent is around 1/3, while for helical turbulence it is between 0.6 and 2/3. In the statistically steady state, the rate of the energy conversion from kinetic into magnetic by the dynamo must be equal to the Joule dissipation rate. We emphasize that for both small-scale and large-scale dynamos, the efficiency of energy conversion depends sensitively on the magnetic Prandtl number, and thus on the microphysical dissipation process. To understand this behavior, we also study shell models of turbulence and one-dimensional passive and active scalar models. We conclude that the magnetic Prandtl number dependence is qualitatively best reproduced in the one-dimensional model as a result of dissipation via localized Alfven kinks.

Magnetic flux concentrations in a polytropic atmosphere

Strongly stratified hydromagnetic turbulence has recently been identified as a candidate for explaining the spontaneous formation of magnetic flux concentrations by the negative effective magnetic pressure instability (NEMPI). Much of this work has been done for isothermal layers for which the density scale height is constant throughout. We now study the validity of earlier conclusions about the size and growth rate of magnetic structures in the case of polytropic layers, which scale height decreases sharply towards the surface. To allow for a continuous transition from isothermal to polytropic layers, we employ a generalization of the exponential function known as the q-exponential. Now, the top of the polytropic layer shifts with the polytropic index such that the scale height at some reference height is always the same. We use both mean-field and direct numerical simulations of forced stratified turbulence to determine the resulting flux concentrations. Magnetic structures begin to form at a depth where magnetic field strength is about 3-4% the local equipartition field strength with respect to the turbulent kinetic energy. Unlike the isothermal case where stronger fields can give rise to magnetic flux concentrations at larger depths, in the polytropic one the growth rates decreases for structures deeper down. For vertical fields, magnetic structures of super-equipartition strengths are formed because such fields survive downward advection, unlike NEMPI under horizontal magnetic fields. The horizontal cross-section of such structures is approximately circular. Results based on isothermal models can be applied locally to polytropic layers. For vertical fields, magnetic flux concentrations of super-equipartition strengths form, which supports suggestions that sunspot formation might be a shallow phenomenon.

Accretion in giant planet circumplanetary discs

During the final growth phase of giant planets, accretion is thought to be controlled by a surrounding circumplanetary disk. Current astrophysical accretion disk models rely on hydromagnetic turbulence or gravitoturbulence as the source of effective viscosity within the disk. However, the magnetically-coupled accreting region in these models is so limited that the disk may not support inflow at all radii, or at the required rate. Here, we examine the conditions needed for self-consistent accretion, in which the disk is susceptible to accretion driven by magnetic fields or gravitational instability. We model the disk as a Shakura-Sunyaev $\alpha$ disk and calculate the level of ionisation, the strength of coupling between the field and disk using Ohmic, Hall and Ambipolar diffusevities for both an MRI and vertical field, and the strength of gravitational instability. We find that the standard constant-$\alpha$ disk is only coupled to the field by thermal ionisation within $30\,R_J$ with strong magnetic diffusivity prohibiting accretion through the bulk of the midplane. In light of the failure of the constant-$\alpha$ disk to produce accretion consistent with its viscosity we drop the assumption of constant-$\alpha$ and present an alternate model in which $\alpha$ varies radially according to the level magnetic turbulence or gravitoturbulence. We find that a vertical field may drive accretion across the entire disk, whereas MRI can drive accretion out to $\sim200\,R_J$, beyond which Toomre's $Q=1$ and gravitoturbulence dominates. The disks are relatively hot ($T\gtrsim800\,$K), and consequently massive ($M_{\text{disk}}\sim0.5\,M_J$).

Mean-field and direct numerical simulations of magnetic flux concentrations from vertical field

Strongly stratified hydromagnetic turbulence has previously been found to produce magnetic flux concentrations if the domain is large enough compared with the size of turbulent eddies. Mean-field simulations (MFS) using parameterizations of the Reynolds and Maxwell stresses show a negative effective magnetic pressure instability and have been able to reproduce many aspects of direct numerical simulations (DNS) regarding the growth rate of this large-scale instability, shape of the resulting magnetic structures, and their height as a function of magnetic field strength. Unlike the case of an imposed horizontal field, for a vertical one, magnetic flux concentrations of equipartition strength with the turbulence can be reached. This results in magnetic spots that are reminiscent of sunspots. Here we want to find out under what conditions magnetic flux concentrations with vertical field occur and what their internal structure is. We use a combination of MFS, DNS, and implicit large-eddy simulations to characterize the resulting magnetic flux concentrations in forced isothermal turbulence with an imposed vertical magnetic field. We confirm earlier results that in the kinematic stage of the large-scale instability the horizontal wavelength of structures is about 10 times the density scale height. At later times, even larger structures are being produced in a fashion similar to inverse spectral transfer in helically driven turbulence. Using turbulence simulations, we find that magnetic flux concentrations occur for different values of the Mach number between 0.1 and 0.7. DNS and MFS show magnetic flux tubes with mean-field energies comparable to the turbulent kinetic energy. The resulting vertical magnetic flux tubes are being confined by downflows along the tubes and corresponding inflow from the sides, which keep the field concentrated.

Astrophysical Hydromagnetic Turbulence

Recent progress in astrophysical hydromagnetic turbulence is being reviewed. The physical ideas behind the now widely accepted Goldreich--Sridhar model and its extension to compressible magnetohydrodynamic turbulence are introduced. Implications for cosmic ray diffusion and acceleration is being discussed. Dynamo-generated magnetic fields with and without helicity are contrasted against each other. Certain turbulent transport processes are being modified and often suppressed by anisotropy and inhomogeneities of the turbulence, while others are being produced by such properties, which can lead to new large-scale instabilities of the turbulent medium. Applications of various such processes to astrophysical systems are being considered.

DYNAMO EFFECTS IN MAGNETOROTATIONAL TURBULENCE WITH FINITE THERMAL DIFFUSIVITY

We investigate the saturation level of hydromagnetic turbulence driven by the magnetorotational instability in the case of vanishing net flux. Motivated by a recent paper of Bodo, Cattaneo, Mignone, & Rossi, we here focus on the case of a non-isothermal equation of state with constant thermal diffusivity. The central aim of the paper is to complement the previous result with closure parameters for mean-field dynamo models, and to test the hypothesis that the dynamo is affected by the mode of heat transport. We perform computer simulations of local shearing-box models of stratified accretion disks with approximate treatment of radiative heat transport, which is modeled via thermal conduction. We study the effect of varying the (constant) thermal diffusivity, and apply different vertical boundary conditions. In the case of impenetrable vertical boundaries, we confirm the transition from mainly conductive to mainly convective vertical heat transport below a critical thermal diffusivity. This transition is however much less dramatic when more natural outflow boundary conditions are applied. Similarly, the enhancement of magnetic activity in this case is less pronounced. Nevertheless, heating via turbulent dissipation determines the thermodynamic structure of accretion disks, and clearly affects the properties of the related dynamo. This effect may however have been overestimated in previous work, and a careful study of the role played by boundaries will be required.

Evolution of primordial magnetic fields from phase transitions

We consider the evolution of primordial magnetic fields generated during cosmological, electroweak or QCD, phase transitions. We assume that the magnetic field generation can be described as an injection of magnetic energy to cosmological plasma at a given scale determined by the moment of magnetic field generation. A high Reynolds number ensures strong coupling between magnetic field and fluid motions. The subsequent evolution of the magnetic field is governed by decaying hydromagnetic turbulence. Both our numerical simulations and a phenomenological description allow us to recover "universal" laws for the decay of magnetic energy and the growth of magnetic correlation length in the turbulent (low viscosity) regime. In particular, we show that during the radiation dominated epoch, energy and correlation length of non-helical magnetic fields scale as conformal time to the powers -1/2 and +1/2, respectively. For helical magnetic fields, energy and correlation length scale as conformal time to the powers -1/3 and +2/3, respectively. The universal decay law of the magnetic field implies that the strength of magnetic field generated during the QCD phase transition could reach $\sim 10^{-9}$\,G with the present day correlation length $\sim 50$ kpc. The fields generated at the electroweak phase transition could be as strong as $\sim 10^{-10}$ G with correlation lengths reaching $\sim 0.3$\,kpc. These values of the magnetic fields are consistent with the lower bounds of the extragalactic magnetic fields.

Spontaneous Formation of Magnetic Flux Concentrations in Stratified Turbulence

The negative effective magnetic pressure instability discovered recently in direct numerical simulations (DNS) may play a crucial role in the formation of sunspots and active regions in the Sun and stars. This instability is caused by a negative contribution of turbulence to the effective mean Lorentz force (the sum of turbulent and non-turbulent contributions) and results in formation of large-scale inhomogeneous magnetic structures from initial uniform magnetic field. Earlier investigations of this instability in DNS of stably stratified, externally forced, isothermal hydromagnetic turbulence in the regime of large plasma beta are now extended into the regime of larger scale separation ratios where the number of turbulent eddies in the computational domain is about 30. Strong spontaneous formation of large-scale magnetic structures is seen even without performing any spatial averaging. These structures encompass many turbulent eddies. The characteristic time of the instability is comparable to the turbulent diffusion time, L^2/eta_t, where eta_t is the turbulent diffusivity and L is the scale of the domain. DNS are used to confirm that the effective magnetic pressure does indeed become negative for magnetic field strengths below the equipartition field. The dependence of the effective magnetic pressure on the field strength is characterized by fit parameters that seem to show convergence for larger values of the magnetic Reynolds number.

DETECTION OF NEGATIVE EFFECTIVE MAGNETIC PRESSURE INSTABILITY IN TURBULENCE SIMULATIONS

We present the first demonstration of the negative effective magnetic pressure instability in direct numerical simulations of stably stratified, externally forced, isothermal hydromagnetic turbulence in the regime of large plasma beta. By the action of this instability, an initially uniform horizontal magnetic field forms flux concentrations whose scale is large compared to the turbulent scale. We further show that the magnetic energy of these large-scale structures is only weakly dependent on the magnetic Reynolds number, provided its value is large enough for the instability to be excited. Our results support earlier mean-field calculations and analytic work which identified this instability. Applications to the formation of active regions in the Sun are discussed.

A POSSIBLE APPROACH TO THREE-DIMENSIONAL COSMIC-RAY PROPAGATION IN THE GALAXY. IV. ELECTRONS AND ELECTRON-INDUCED γ-RAYS

Based on the diffusion-halo model for cosmic-ray (CR) propagation, including stochastic reacceleration due to collisions with hydromagnetic turbulence, we study the behavior of the electron component and the diffuse $\gamma$-rays (D$\gamma$'s) induced by them. The galactic parameters appearing in these studies are essentially the same as those appearing in the hadronic CR components, while we additionally need information on the interstellar radiation field, taking into account dependences on both the photon energy, $E_{\scriptsize {ph}}$, and the position, $\vct{r}$. We compare our numerical results with the data on hadrons, electrons and D$\gamma$'s, including the most recent results from FERMI, which gives two remarkable results; 1) the electron spectrum falls with energy as $E_e^{-3}$ up to 1\,TeV, and does not exhibit prominent spectral features around 500\,GeV, in contrast to the dramatic excess appearing in both ATIC and PPB-BETS spectra, and 2) the EGRET GeV-excess in the D$\gamma$ spectrum is due neither to an astronomical origin (much harder CR spectrum in the galactic center) nor a cosmological one (dark matter annihilation or decay), but due to an instrumental problem. In the present paper, however, we focus our interest rather conservatively upon the internal relation between these three components, using {\it common} galactic parameters. We find that they are in reasonable harmony with each other within both the theoretical and experimental uncertainties, apart from the electron-anomaly problem, while some enhancement of D$\gamma$'s appears in the high galactic latitude with $|b| > 60^\circ$ in the GeV region.

On magnetoacoustic-gravity-inertial (MAGI) waves - I. Generation, propagation, dissipation and radiation

The general equations of fluid mechanics under gravity, rotation and external magnetic fields are used to study the generation, propagation and radiation of magnetoacoustic-gravity-inertial (MAGI) waves, coupling the four restoring forces: pressure, gravity, magnetic field and Coriolis-centrifugal forces. The mean state is assumed to be an ionized rotating atmosphere, and the elimination among the linear, non-dissipative perturbations leads to a matrix MAGI wave operator applying to the velocity perturbation vector, which describes acoustic, magnetic, internal and inertial waves and all their couplings. The non-linear and viscous, thermal and resistive dissipative terms are retained, to model the sources of MAGI waves, which consist of hydrodynamic, hydromagnetic, hydrogravity and hydrorotation tensors, representing MAGI wave generation by (i) stratified hydromagnetic turbulence; (ii) ionized fluid inhomogeneities and (iii) heat dissipation by fluid viscosity and electrical resistance. The energy flux of MAGI wave sources is used in Part II to estimate the radiative losses of the Sun, and the luminosities of a variety of stars, including magnetic and non-magnetic and rotating and non-rotating, supporting the idea that energy (and mass) transport in stellar atmospheres is due to MAGI waves.

Supernova-driven interstellar turbulence and the galactic dynamo

AbstractThe fractal shape and multi-component nature of the interstellar medium together with its vast range of dynamical scales provides one of the great challenges in theoretical and numerical astrophysics. Here we will review recent progress in the direct modelling of interstellar hydromagnetic turbulence, focusing on the role of energy injection by supernova explosions. The implications for dynamo theory will be discussed in the context of the mean-field approach.Results obtained with the test field-method are confronted with analytical predictions and estimates from quasilinear theory. The simulation results enforce the classical understanding of a turbulent Galactic dynamo and, more importantly, yield new quantitative insights. The derived scaling relations enable confident global mean-field modelling.

Using gamma-ray burst prompt emission to probe relativistic shock acceleration

It is widely accepted that the prompt transient signal in the 10 keV–10 GeV band from gamma-ray bursts (GRBs) arises from multiple shocks internal to the ultra-relativistic expansion. The detailed understanding of the dissipation and accompanying acceleration at these shocks is a currently topical subject. This paper explores the relationship between GRB prompt emission spectra and the electron (or ion) acceleration properties at the relativistic shocks that pertain to GRB models. The focus is on the array of possible high-energy power-law indices in accelerated populations, highlighting how spectra above 1 MeV can probe the field obliquity in GRB internal shocks, and the character of hydromagnetic turbulence in their environs. It is emphasized that diffusive shock acceleration theory generates no canonical spectrum at relativistic MHD discontinuities. This diversity is commensurate with the significant range of spectral indices discerned in prompt burst emission. Such system diagnostics are now being enhanced by the broad-band spectral coverage of bursts by the Fermi Gamma-Ray Space Telescope; while the Gamma-Ray Burst Monitor (GBM) provides key diagnostics on the lower energy portions of the particle population, the focus here is on constraints in the non-thermal, power-law regime of the particle distribution that are provided by the Large Area Telescope (LAT).

Magnetic field evolution in simulations with Euler potentials

Using two- and three-dimensional hydromagnetic simulations for a range of different flows, including laminar and turbulent ones, it is shown that solutions expressing the field in terms of Euler potentials (EP) are in general incorrect if the EP are evolved with an artificial diffusion term. In three dimensions, standard methods using the magnetic vector potential are found to permit dynamo action when the EP give decaying solutions. With an imposed field, the EP method yields excessive power at small scales. This effect is more exaggerated in the dynamic case, suggesting an unrealistically reduced feedback from the Lorentz force. The EP approach agrees with standard methods only at early times when magnetic diffusivity did not have time to act. It is demonstrated that the usage of EP with even a small artificial magnetic diffusivity does not converge to a proper solution of hydromagnetic turbulence. The source of this disagreement is not connected with magnetic helicity or the three-dimensionality of the magnetic field, but is simply due to the fact that the non-linear representation of the magnetic field in terms of EP that depend on the same coordinates is incompatible with the linear diffusion operator in the induction equation.

Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models

We prove that smooth solutions of nonideal (viscous and resistive) incompressible magnetohydrodynamic (MHD) equations satisfy a stochastic law of flux conservation. This property implies that the magnetic flux through a surface is equal to the average of the magnetic fluxes through an ensemble of surfaces advected backward in time by the plasma velocity perturbed with a random white noise. Our result is an analog of the well-known Alfvén theorem of ideal MHD and is valid for any value of the magnetic Prandtl number. A second stochastic conservation law is shown to hold at unit Prandtl number, a random version of the generalized Kelvin theorem derived by Bekenstein and Oron for ideal MHD. These stochastic conservation laws are not only shown to be consequences of the nonideal MHD equations but are proved in fact to be equivalent to those equations. We derive similar results for two more refined hydromagnetic models, Hall MHD and the two-fluid plasma model, still assuming incompressible velocities and isotropic transport coefficients. Finally, we use these results to discuss briefly the infinite-Reynolds-number limit of hydromagnetic turbulence and to support the conjecture that flux conservation remains stochastic in that limit.

LARGE-SCALE DYNAMOS AT LOW MAGNETIC PRANDTL NUMBERS

Using direct simulations of hydromagnetic turbulence driven by random polarized waves it is shown that dynamo action is possible over a wide range of magnetic Prandtl numbers from 10^-3 to 1. Triply periodic boundary conditions are being used. In the final saturated state the resulting magnetic field has a large-scale component of Beltrami type. For the kinematic phase, growth rates have been determined for magnetic Prandtl numbers between 0.01 and 1, but only the case with the smallest magnetic Prandtl number shows large-scale magnetic fields. It is less organized than in the nonlinear stage. For small magnetic Prandtl numbers the growth rates are comparable to those calculated from an alpha squared mean-field dynamo. In the linear regime the magnetic helicity spectrum has a short inertial range compatible with a -5/3 power law, while in the nonlinear regime it is the current helicity whose spectrum may be compatible with such a law. In the saturated case, the spectral magnetic energy in the inertial range is in slight excess over the spectral kinetic energy, although for small magnetic Prandtl numbers the magnetic energy spectrum reaches its resistive cut off wavenumber more quickly. The viscous energy dissipation declines with the square root of the magnetic Prandtl number, which implies that most of the energy is dissipated via Joule heat.