Growth Of Magnetic Energy Research Articles

Energy transfers and magnetic energy growth in small-scale dynamo

In this letter we investigate the dynamics of magnetic energy growth in small-scale dynamo by studying energy transfers, mainly energy fluxes and shell-to-shell energy transfers. We perform dynamo simulations for the magnetic Prandtl number on 10243 grid using the pseudospectral method. We demonstrate that the magnetic energy growth is caused by nonlocal energy transfers from the large-scale or forcing-scale velocity field to small-scale magnetic field. The peak of these energy transfers moves towards lower wave numbers as dynamo evolves, which is the reason why the integral scale of the magnetic field increases with time. The energy transfers U2U (velocity to velocity) and B2B (magnetic to magnetic) are forward and local.

The small-scale dynamo: breaking universality at high Mach numbers

The small-scale dynamo plays a substantial role in magnetizing the Universe under a large range of conditions, including subsonic turbulence at low Mach numbers, highly supersonic turbulence at high Mach numbers and a large range of magnetic Prandtl numbers Pm, i.e. the ratio of kinetic viscosity to magnetic resistivity. Low Mach numbers may, in particular, lead to the well-known, incompressible Kolmogorov turbulence, while for high Mach numbers, we are in the highly compressible regime, thus close to Burgers turbulence. In this paper, we explore whether in this large range of conditions, universal behavior can be expected. Our starting point is previous investigations in the kinematic regime. Here, analytic studies based on the Kazantsev model have shown that the behavior of the dynamo depends significantly on Pm and the type of turbulence, and numerical simulations indicate a strong dependence of the growth rate on the Mach number of the flow. Once the magnetic field saturates on the current amplification scale, backreactions occur and the growth is shifted to the next-larger scale. We employ a Fokker–Planck model to calculate the magnetic field amplification during the nonlinear regime, and find a resulting power-law growth that depends on the type of turbulence invoked. For Kolmogorov turbulence, we confirm previous results suggesting a linear growth of magnetic energy. For more general turbulent spectra, where the turbulent velocity scales with the characteristic length scale as uℓ∝ℓϑ, we find that the magnetic energy grows as (t/Ted)2ϑ/(1−ϑ), with t being the time coordinate and Ted the eddy-turnover time on the forcing scale of turbulence. For Burgers turbulence, ϑ = 1/2, quadratic rather than linear growth may thus be expected, as the spectral energy increases from smaller to larger scales more rapidly. The quadratic growth is due to the initially smaller growth rates obtained for Burgers turbulence. Similarly, we show that the characteristic length scale of the magnetic field grows as t1/(1−ϑ) in the general case, implying t3/2 for Kolmogorov and t2 for Burgers turbulence. Overall, we find that high Mach numbers, as typically associated with steep spectra of turbulence, may break the previously postulated universality, and introduce a dependence on the environment also in the nonlinear regime.

Optimization of the Magnetic Dynamo

In stars and planets, magnetic fields are believed to originate from the motion of electrically conducting fluids in their interior, through a process known as the dynamo mechanism. In this Letter, an optimization procedure is used to simultaneously address two fundamental questions of dynamo theory: "Which velocity field leads to the most magnetic energy growth?" and "How large does the velocity need to be relative to magnetic diffusion?" In general, this requires optimization over the full space of continuous solenoidal velocity fields possible within the geometry. Here the case of a periodic box is considered. Measuring the strength of the flow with the root-mean-square amplitude, an optimal velocity field is shown to exist, but without limitation on the strain rate, optimization is prone to divergence. Measuring the flow in terms of its associated dissipation leads to the identification of a single optimal at the critical magnetic Reynolds number necessary for a dynamo. This magnetic Reynolds number is found to be only 15% higher than that necessary for transient growth of the magnetic field.

NUMERICAL SIMULATIONS OF DRIVEN RELATIVISTIC MAGNETOHYDRODYNAMIC TURBULENCE

A wide variety of astrophysical phenomena involve the flow of turbulent magnetized gas with relativistic velocity or energy density. Examples include gamma-ray bursts, active galactic nuclei, pulsars, magnetars, micro-quasars, merging neutron stars, X-ray binaries, some supernovae, and the early universe. In order to elucidate the basic properties of the relativistic magnetohydrodynamical (RMHD) turbulence present in these systems, we present results from numerical simulations of fully developed driven turbulence in a relativistically warm, weakly magnetized and mildly compressible ideal fluid. We have evolved the RMHD equations for many dynamical times on a uniform grid with 1024^3 zones using a high order Godunov code. We observe the growth of magnetic energy from a seed field through saturation at about 1% of the total fluid energy. We compute the power spectrum of velocity and density-weighted velocity and conclude that the inertial scaling is consistent with a slope of -5/3. We compute the longitudinal and transverse velocity structure functions of order p up to 11, and discuss their possible deviation from the expected scaling for non-relativistic media. We also compute the scale-dependent distortion of coherent velocity structures with respect to the local magnetic field, finding a weaker scale dependence than is expected for incompressible non-relativistic flows with a strong mean field.

Core-collapse supernovae: Magnetorotational explosions and jet formation

Core-collapse supernovae are accompanied by formation of neutron stars. The gravitational energy is transformed into the energy of the explosion, observed as SN II, SN Ib,c type supernovae. We present results of 2D MHD simulations, where the source of energy is rotation and the magnetic field serves as a “transition belt” for the transformation of the rotation energy into the energy of the explosion. The toroidal part of the magnetic energy initially grows linearly with time due to differential rotation. When the twisted toroidal component strongly exceeds the poloidal field, magneto-rotational instability develops, leading to a drastic acceleration in the growth of magnetic energy. Finally, a fast MHD shock is formed, producing a supernova explosion. A mildly collimated jet is produced for the dipolelike type of the initial field.

Core-Collapse Supernovae: Magnetorotational Explosions and Jet Formation

Core-collapse supernovae are accompanied by formation of neutron stars. The gravitational energy is transformed into the energy of the explosion, observed as SN II, SN Ib,c type supernovae. We present results of 2D MHD simulations, where the source of energy is rotation and the magnetic field serves as a “transition belt” for the transformation of the rotation energy into the energy of the explosion. The toroidal part of the magnetic energy initially grows linearly with time due to differential rotation. When the twisted toroidal component strongly exceeds the poloidal field, magneto-rotational instability develops, leading to a drastic acceleration in the growth of magnetic energy. Finally, a fast MHD shock is formed, producing a supernova explosion. A mildly collimated jet is produced for the dipolelike type of the initial field.

Inverse cascades andαeffect at a low magnetic Prandtl number

Dynamo action in a fully helical Arn'old-Beltrami-Childress flow is studied using both direct numerical simulations and subgrid modeling. Sufficient scale separation is given in order to allow for large-scale magnetic energy buildup. Growth of magnetic energy obtains down to a magnetic Prandtl number P(M) = R(M)/R(V) close to 0.005, where R(V) and R(M) are the kinetic and magnetic Reynolds numbers. The critical magnetic Reynolds number for dynamo action R(M)(c) seems to saturate at values close to 20. Detailed studies of the dependence of the amplitude of the saturated magnetic energy with P(M) are presented. When P(M) is decreased, numerical experiments are conducted with either R(V) or R(M) kept constant. In the former case, the ratio of magnetic to kinetic energy saturates to a value slightly below unity as P(M) decreases. Examination of energy spectra and structures in real space reveals that quenching of the velocity by a large-scale magnetic field takes place, with an inverse cascade of magnetic helicity and a force-free field at large scale in the saturated regime.

Transient magnetic energy growth in spherical stationary flows

The non-normality associated with the induction equation may lead to subcritical growth of magnetic field structures even if all linear eigenmodes decay. We compute the magnitude of these transient effects for a collection of predominantly axisymmetric stationary spherical flows and find without exception the dominance of axisymmetric field growth above all other symmetries. The transient growth is robust under small flow perturbations and can be understood by simple physical mechanisms: either field line shearing or stretching. Magnetic energy amplification of is possible at magnetic Reynolds numbers of , and such effects could therefore lead the system from a principally non-magnetic state into one where the magnetic field plays a significant role in the dynamics.

Saturation properties of the Archontis dynamo

This paper discusses the generation and subsequent non-linear limiting of magnetic fields by motions in a periodic flow driven by a force whose components are proportional to (sin z, sin x, sin y). This problem was originally studied by Archontis; the purpose of the present work is to remove certain complications present in the original model in order to understand better the underlying physical mechanism which limits the total magnetic energy growth. At high Reynolds numbers the resulting dynamos end up with almost equal total magnetic and kinetic energies, and thus yield fields strong enough to be astrophysically relevant. Until now, the existence of such dynamos has been doubted, so this demonstration of an example appears very important. At least up to Reynolds numbers of 800, these solutions are laminar and attracting. We then present an argument showing that any stationary, incompressible dynamo can be used to create a family of solutions with an arbitrary ratio of magnetic to kinetic energies in the limit of large Reynolds numbers. The stability of such solutions is also discussed.

Dynamo simulations in a spherical shell of ideal gas using a high-order cartesian magnetohydrodynamics code

Most numerical simulations of the geodynamo are cast in spherical geometry, using a spherical harmonic representation for lateral variations, and an expansion in Chebyshev polynomials, or discretisation, in radius. A number of research groups have produced three-dimensional, time-dependent, self-consistent simulations of the geodynamo using this pseudospectral methodology. Computational limitations place a practical bound on the parameter regime that can be explored in this context, however, and Earth-like values currently remain several orders of magnitude out of reach. For the spherically pseudospectral codes, the absence of a fast Legendre transform is a strong factor contributing to this limitation. As a first step toward alternative computational methods for geodynamo modelling, we adapt a pre-existing, efficiently parallelised magnetohydrodynamics (MHD) code, to model spherical shell problems. The Pencil-Code is a Cartesian code that uses sixth-order finite differences, applied to ‘pencils’ (i.e. array sections) in the x direction in a cache-efficient way. The domain is tiled in the y and z directions, with the communication of boundary elements handled by Message Passing Interface (MPI). Time stepping is via a third order Runge-Kutta method. The code’s modular structure allows a flexible selection of various physical processes and variables, making it easily adaptable for many types of compressible MHD problems. We demonstrate dynamo action driven by thermal convection in a spherical shell of compressible ideal gas, for comparison with the ideal gas dynamos of Kageyama and Sato [Kageyama, A., Sato, T., the Complexity Simulation Group, 1995. Computer simulation of a magnetohydrodynamic dynamo. II. Phys. Fluids, B 2, 2793–2805]. As with the previous results, our solutions are characterised by steady state thermal convection with axial columnar cells, and exponential growth and subsequent saturation of magnetic energy. Minor differences in the details of the benchmark and current simulations are attributed to differences in the computational methods and geometry of the domains involved. We discuss the further modifications still needed for a fuller adaptation to geodynamo problems.

Amplification of Interstellar Magnetic Fields and Turbulent Mixing by Supernova‐driven Turbulence. II. The Role of Dynamical Chaos

In this paper we further advance the study of magnetic field amplification in the interstellar medium that we started in 2004. We show that the flux growth rate is comparable to the rate of magnetic energy growth found in the earlier paper. We also demonstrate the role of intermittency in field amplification. The density shows a double-peaked PDF, consistent with the cooling curve that was used. The PDF of the magnetic field shows a high-end tail, providing a telltale signature of the operation of the small-scale dynamo. The magnetic field strength correlates with the density as ~ ρα, with α = 0.386. As a result, the field amplification takes place more vigorously in the lower temperature, denser gas. The Lagrangian chaos in the simulated turbulent flows is studied in substantial detail. It is shown that the stretching rate of material lines, as well as the Lyapunov exponents, can be used to gain important insights into the growth of magnetic field. The cancellation exponent for the small-scale supernova-driven dynamo is derived, and it is shown that constructive folding of field lines in the dynamo is very inefficient. We also show that our Lagrangian approach can yield actual measures of the turbulent diffusivity in the simulated ISM. The turbulent diffusivity provides insights into the mixing of elements from supernova ejecta on macroscopic scales. The high rates of line stretching in interstellar turbulence suggests that the eventual diffusion of elements at the molecular level is very efficient. Many of the diagnostics of turbulence that are presented here can be used to make direct connections between simulations and observations.

Study of neoclassical tearing modes based on a reduced MHD model in cylindrical geometry

Neoclassical tearing modes (NTMs) are examined with a reduced MHD model including the bootstrap current as a simple model, JBS ∝ − dP/dr, in cylindrical tokamaks. It is found that linear instabilities due to the perturbation of bootstrap current appear for large JBS cases with large parallel thermal diffusivity, χ∥, under small perpendicular thermal diffusivity, χ⊥, in the pressure evolution equation. Here, other dissipations such as resistivity and viscosity are also finite. Although the linear growth rate of this unstable mode is small, the growth of magnetic energy of this unstable mode is usually accelerated after the saturation of kinetic energy. Its nonlinear behaviour shown by the evolution of magnetic islands is consistent with the standard nonlinear NTM theory.

Magnetorotational supernovae

We present the results of 2D simulations of the magnetorotational model of a supernova explosion. After the core collapse the core consists of rapidly a rotating proto-neutron star and a differentially rotating envelope. The toroidal part of the magnetic energy generated by the differential rotation grows linearly with time at the initial stage of the evolution of the magnetic field. The linear growth of the toroidal magnetic field is terminated by the development of magnetohydrodynamic instability, leading to drastic acceleration in the growth of magnetic energy. At the moment when the magnetic pressure becomes comparable with the gas pressure at the periphery of the proto-neutron star $\sim 10-15$km from the star centre the MHD compression wave appears and goes through the envelope of the collapsed iron core. It transforms soon to the fast MHD shock and produces a supernova explosion. Our simulations give the energy of the explosion $0.6\cdot 10^{51}$ ergs. The amount of the mass ejected by the explosion is $\sim 0.14M_\odot$. The implicit numerical method, based on the Lagrangian triangular grid of variable structure, was used for the simulations.

Amplification of Interstellar Magnetic Fields by Supernova‐driven Turbulence

Several lines of evidence suggest that magnetic fields grow rapidly in protogalactic and galactic environments. However, mean field dynamo theory has always suggested that the magnetic fields grow rather slowly, taking of order a Hubble time to reach the observed values. The theoretical difficulties only become worse when the system has a high magnetic Reynolds number, as is the case for galactic and protogalactic environments. The discrepancy can be reconciled if fast processes for amplifying the magnetic field could operate. Following the 2001 work of Balsara and coworkers, we show that an interstellar medium that is dominated by realistic energy input from supernova explosions will naturally become a strongly turbulent medium with large positive and negative values of the kinetic helicity. Even though the medium is driven by compressible motions, the kinetic energy in this high Mach number flow is mainly concentrated in solenoidal rather than compressible motions. These results stem from the interaction of strong shocks with each other and with the interstellar turbulence they self-consistently generate in our model. Moreover, this interaction also generates large kinetic helicities of either sign. The turbulent flow that we model has two other characteristics of a fast dynamo: magnetic energy growth independent of scale and a growth time that is comparable to the eddy turnover time. This linear phase of growth permits the field to grow rapidly until the magnetic energy reaches about 1% of the kinetic energy. At that stage, other astrophysical processes for producing magnetic fields can take over. Energetics, power spectra, statistics, and structures of the turbulent flow are studied here. Shock-turbulence interaction is shown to be a very general mechanism for helicity generation and magnetic field amplification, with applicability to damped Ly? systems, protogalaxies, the Galaxy, starburst galaxies, the intracluster medium, and molecular clouds.

Simulations of the Small‐Scale Turbulent Dynamo

We report the results of an extensive numerical study of the small-scale turbulent dynamo. The primary focus is on the case of large magnetic Prandtl numbers Prm, which is relevant for hot low-density astrophysical plasmas. A Prm parameter scan is given for the model case of viscosity-dominated (low Reynolds number) turbulence. We concentrate on three topics: magnetic energy spectra and saturation levels, the structure of the magnetic field lines, and intermittency of the field strength distribution. The main results are as follows: (1) the folded structure of the field (direction reversals at the resistive scale, field lines curved at the scale of the flow) persists from the kinematic to the nonlinear regime; (2) the field distribution is self-similar and appears to be lognormal during the kinematic regime and exponential in the saturated state; and (3) the bulk of the magnetic energy is at the resistive scale in the kinematic regime and remains there after saturation, although the magnetic energy spectrum becomes much shallower. We propose an analytical model of saturation based on the idea of partial two-dimensionalization of the velocity gradients with respect to the local direction of the magnetic folds. The model-predicted saturated spectra are in excellent agreement with numerical results. Comparisons with large-Re, moderate-Prm runs are carried out to confirm the relevance of these results and to test heuristic scenarios of dynamo saturation. New features at large Re are elongation of the folds in the nonlinear regime from the viscous scale to the box scale and the presence of an intermediate nonlinear stage of slower than exponential magnetic energy growth accompanied by an increase of the resistive scale and partial suppression of the kinetic energy spectrum in the inertial range. Numerical results for the saturated state do not support scale-by-scale equipartition between magnetic and kinetic energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is proposed. Subject heading gs: magnetic fields — methods: numerical — MHD — plasmas — turbulence

The Nonlinear Magnetic Cascade

The Galactic magnetic field has an energy density comparable to that of the interstellar medium turbulence and a coherence spanning the Galaxy. It is not known if this field was formed before, during, or after the Galaxy. However, it is often assumed to originate from a turbulent dynamo process. We investigate the early stages of a Galactic dynamo when the dynamics is well approximated by homogeneous turbulence. Our simulations show that homogeneous magnetized turbulence with large Prandtl number yields magnetic energy at the small, resistive scale rather than at the Galactic scale. Thus, additional phenomena—perhaps helicity generated from Galactic rotation, stratification, and the differential rotation of the disk—are needed to explain the observed field. We simulate the growth of magnetic energy in forced nonhelical turbulence from an initially weak value until it saturates with the same energy density as the turbulence. When the field is dynamically weak, the simulations agree with the kinematic theory. In the long-term saturated state, the magnetic field is strong enough to modify the turbulence. This is the magnetohydrodynamic (MHD) analog of the Kolmogorov problem for hydrodynamic turbulence. The nature of the back-reaction is to neutralize the net shear (stretching) in small-scale eddies that are less energetic than the magnetic field. Only the forcing-scale eddies remain energetic enough to shear and cascade the magnetic field. The magnetic field at all scales therefore forward-cascades at the forcing timescale through spectrally nonlocal interactions with the forcing-scale eddies. Furthermore, the magnetic field folds into a reduced-tension state where field-line curvature anticorrelates with intensity. Direct consequences of these statements are that the magnetic spectrum is largely independent of viscosity and that the magnetic energy is located at the small, resistive scale.

Dynamic effects on the stretching of the magnetic field by a plasma flow

A key mechanism in the growth of magnetic energy in kinematic dynamos is the stretching of the magnetic field vector by making it point in an unstable direction of the strain matrix. Our objective is to study whether this feature may be maintained in an ideal plasma when also considering the back reaction of the magnetic field upon the flow through the Lorentz force. Several effects occur: in addition to the nonlocal ones exerted by the total pressure, a complex geometry of magnetic field lines decreases the rate of growth of magnetic energy, rotation of the flow enhances it and above all the rate of growth decreases with minus the square of the eigenvalue associated with the magnetic field direction. Thus local dynamics tend to rapidly quench the stretching of the field.

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.

Relaxation processes in a low-order three-dimensional magnetohydrodynamics model

The selective decay and dynamic alignment relaxation theories are used to interpret the time asymptotic behavior of a Galerkin model of three-dimensional (3-D) magnetohydrodynamics (MHD). A large number of simulations are performed that scan a parameter space defined by the rugged ideal invariants: energy, cross helicity, and magnetic helicity. Ranges of the initial parameters are found where one or both of the relaxation theories are needed to describe the time asymptotic properties of the system, as previously found in analogous studies of two-dimensional (2-D) MHD [Ting et al., Phys. Fluids 29, 3261 (1986)]. In many cases, the time asymptotic state can be interpreted as a relaxation to minimum energy. For certain parameter ranges spectral back transfer of cross helicity can lead to growth in velocity-magnetic field correlation [Stribling and Matthaeus, Phys. Fluids B 2, 1979 (1990)]. A simple decay model, based on absolute equilibrium theory, predicts a mapping of initial onto time asymptotic states, and accurately describes the long time behavior of the runs when magnetic helicity is present. We also discuss two processes, operating on time scales shorter than selective decay and dynamic alignment, in which the ratio of kinetic to magnetic energy relaxes to values O(1). The faster of the two takes states initially dominant in magnetic energy to a state of near-equipartition between kinetic and magnetic energy through power law growth of kinetic energy. The other process takes states initially dominant in kinetic energy to the near-equipartitioned state through exponential growth of magnetic energy.

Hyper-resistivity produced by tearing mode turbulence

Tearing mode turbulence produces a hyper-resistivity or effective anomalous electron viscosity. The hyper-resistivity is calculated for the mean magnetic field quasilinearly, and for long-wavelength modes using the direct interaction approximation. The hyper-resistivity accounts for current relaxation in reversed-field pinch experiments, and gives a magnetic fluctuation sealing of S−1/3. It causes enhanced tearing mode growth rates in the turbulent phase of tokamak disruptions. In astrophysics, it limits magnetic energy growth due to the dynamo effect, and may explain rapid reconnection phenomena such as solar flares.